11.4: Cognitive Development in Early Adulthood
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Learning Objectives
By the end of this section, you will be able to:
- Identify the cognitive capabilities that develop during early adulthood
- Describe the developmental changes in social cognition associated with early adulthood
- Describe the contextual and cultural factors that influence the development of early adult cognitive capabilities
Devon finished a pipefitting apprenticeship a year ago and is now living fully on their own for the first time. They have just been offered a position managing a small group of people at work. They are excited to be promoted but anxious about overseeing others. It was challenging enough to prioritize their personal life and budget for food, bills, and rent. Now they have to learn how to prioritize projects, make major decisions at work, and delegate parts of each project to those in their work group, on top of making sure they are able to get their work done.
Some of the cognitive tasks and milestones during early adulthood include taking on more responsibilities, finding employment or a career (which may include attending college or a trade school), attaining abstract thought, increasing awareness of our own limitations and strengths, honing decision-making skills, becoming fully independent (including financially independent), being less swayed by peer pressure, and adjusting to changes in relationships with family. Developing and honing new cognitive skills during this time is essential for solving the kinds of complex problems we will encounter for the rest of our lives. Fortunately, our brain continues to develop in ways that make these tasks easier.
Changes in the Brain during Early Adulthood
Just like the rest of the body, the brain reaches maturity and peak functioning during early adulthood. However, we continue to develop new synapse s—a lifelong process—and myelination of our axons continues, particularly in the prefrontal cortex (Dvorak et al., 2021; Hochberg & Konner, 2020; Kwon et al., 2020; Mills et al., 2021), throughout the lifespan. These changes in the brain within the area of the prefrontal cortex are mostly completed by age 25 (citation), and are associated with improved cognitive functioning, specifically in the efficacy and speed of processing and performance of higher-order functions like planning, organizing, solving problems, and resisting impulses (Hochberg & Konner, 2020; Kwon et al., 2020).
In addition to the development within the prefrontal cortex, the process of synaptic pruning continues until the mid-20’s. We also know from research that , with the prefrontal cortex is the last area where synaptic pruning finally winds down within the brain (Cohen et al., 2016; Dennis et al., 2013). These two processes, final synaptic pruning and the continued myelination of nerve fibers in this area, helps increase the efficiency of signals between neurons within the brain. Instead of processing information locally, the brain begins to connect more distant regions, allowing information to be processed in many areas (Hochberg & Konner, 2020; Taber-Thomas & Pérez-Edgar, 2015). Once these connections have been well established, both impulse control and emotional control are enhanced (Simpson, 2018; Wood et al., 2017). Thus, these brain changes help young adults become better at making decisions, delaying gratification, solving problems, assessing risk, prioritizing, and regulating their emotions. Improvements don’t happen all at once. For example, research conducted by Cohen and colleagues (2016) investigated differences in brain functioning between adolescents (ages 13–17), emerging adults ages 18–21, and ages 22–25. The prefrontal cortices of the 18–21-year-olds functioned more like those of adolescents than of the 22–25-year-olds on a task assessing inhibition of a behavior. These results also support Arnett’s theory that emerging adulthood is a discrete developmental state and involves gradual progression rather than an abrupt transition.
Intersections and Contexts: Brain of Emerging Adults: Impulse Control and Risk-Taking Behaviors
Although people can act impulsively regardless of age, such as calling in sick for a week to attend a once-in-a-lifetime music festival or changing one's choice of college major on a whim, this type of behavior occurs more frequently in emerging adulthood than later in adulthood (Brodbeck et al., 2013). Life experience, or lack of, plays some role in the differences in risk-taking and impulsive behaviors between emerging adults and adults. However, these differences are also likely the result of changes in the frontal lobes and other brain areas (Brodbeck et al., 2013). Changes in the brain help it shift from a priority of developing skills and obtaining knowledge about the world to actually using that knowledge and skills to live as an adult.
Unfortunately, this means that while people are exploring options and making decisions more independently, their brain is still in transition, and thus, people may not always make the best decisions. Within the brain, areas that are involved in impulsive behaviors, including risk-taking and decision-making, are also involved in emotional regulation, including impulse control and emotional control. Research has found that people who had difficulty with emotional regulation, based on clinical tests and self-report are prone to more impulsive behaviors related to substance misuse and gambling (Schreiber et al., 2012). Thus, until refinements within these areas are complete, people may make impulsive decisions that bring immediate rewards instead of making decisions based on long-term rewards (Parr et al., 2022).
Another contributor to impulsivity may be the neurotransmitter dopamine. Children, teens, and adults of different ages have different amounts of dopamine in their brains, with dopamine levels decreasing as we age (Lee & Kim, 2022). Similarly, children, teens, and adults of different ages aren’t all equally good at controlling impulses and making goal-based decisions, with older adults being better at this than other age groups, including younger adults. However, some younger adults appear to have less trouble with this than others, suggesting that individual differences may also be at work. Having less dopamine as we age may help us, along with our increased cognitive skills and personal experiences, to better resist impulsive behaviors (Parr et al., 2022).
Early Adult Cognitive Characteristics
Some of the cognitive abilities that develop or become refined through both physiological changes in the brain and experience in daily life are reflective thought, crystallized intelligence, post-formal thought, and dialectical thinking. Up until now, a lot of our coverage of cognitive development has focused on Piaget’s theory and stages. Piaget believed that formal operational thought was the last stage in our cognitive development, but is this actually the case? Do you currently have the same cognitive skills and reason the same way you did when you were 14? Put another way, do you currently have the same problems and life circumstances you had when you were 14?
Researchers since Piaget have proposed the existence of post-formal thought , a stage beyond formal operations that is more flexible, complex, and capable of recognizing more than one objectively correct answer to problems (Kincheloe & Steinberg, 1993; Sinnott, 1998; Griffin et al., 2009). Though adolescents in the formal operational stage can easily think hypothetically, they lack experience with the world and are often unable to consider as many possibilities as adults can. Adults are better able to predict likely outcomes or consequences, combining abstract thought and logic with intuition and life experience. This ability to think of potential outcomes is one of the hallmarks of post-formal thought, and it extends Piaget’s ideas about formal operations to the unique skills and abilities developed during emerging adulthood and adulthood. Thus, it helps people think critically, accept contradictions and ambiguity, and understand situations they’ve never experienced. Table 11.4 shows some key characteristics of post-formal thought and how they may be applied in real life.
| Cognitive Characteristic | What It Does | Example |
|---|---|---|
| Flexible | Aids our ability to transfer between forms of logic, such as between abstract and concrete thought. | A person can think concretely about their favorite video game and then abstractly about how video games can be used for education or stress relief. |
| Helpful in defining problems | Allows us to describe an issue. | Students in a group project might define conflicts they have as miscommunication or a personality clash. |
| Practical | Points us to pragmatic solutions. | Someone weighing two job offers can evaluate the practical aspects of each one (e.g., commute time, work hours, benefits) to decide which one will better meet their goals. |
| Open to multiple solutions | Helps us acknowledge that most problems have more than one cause and solution and that people facing the same problem may have different goals. | A marriage counselor has to consider their clients’ individual situations and not take a “one size fits all” approach. |
| Open to paradox | Shows us that sometimes a situation that seems contradictory may work. | A student doing a term paper spends time writing an outline and researching sources before actually starting to write, enabling them to finish more efficiently than a classmate who just starts writing with no plan in mind. |
In young adulthood, we continue to use many of the cognitive abilities we acquired earlier in life. Skills like memory, perspective taking, and problem solving are still important, although we may use them for different tasks. For example, we no longer just use memory to recall information—we also use it to apply information, remembering not only facts but also how those facts can be used. Nurses must know more than what different medications do; they also have to know how medications interact, how and when they should be administered, and what a typical dosage is. This allows them to detect possible interactions and errors and provide better patient care. We also use existing problem-solving skills to address challenges of adult life, such as what to do in case of a flat tire or how to go about finding an internship for the summer.
During early adulthood, people increase their problem-solving and decision-making skills, become aware that the world is full of gray areas, and recognize that little in life is absolute. They’re better able to see that historical, cultural, and social factors play a role in events, and that different people might view the same event differently due to these factors. Thus they gain a deeper understanding of relativism , or the idea that cultural standards, values, and ethics are a product of the time and cultural context within which they developed, and what is proper, relevant, or true for one individual may not be for another. They also increase their knowledge base, gain more experience with the world and with interacting with others, become more flexible, and hone skills such as planning, prioritizing, strategizing, time management, self-control, and emotional regulation. Table 11.5 lists some cognitive skills and how they develop during early adulthood. Several types of cognitive skills improve or peak at this life stage.
| Cognitive Skill | What Happens in Early Adulthood | Example of Relevant Tasks |
|---|---|---|
| Fluid intelligence | Peaks in our 20s | A psychology student can interpret statistical information to understand a journal article. |
| Crystallized intelligence | Increases until our 40s, when it peaks | An artist can use her knowledge of different sculpture materials to determine the best one to use. |
| Working memory | Improves during our 20s and peaks around age 30 | A nurse can calculate drug dosages in his head. |
| Inhibitory control | Peaks in our 20s | A salesclerk can inhibit their impulse to argue with an unhappy customer. |
| Vocabulary | Increases until late adulthood | A therapy client can find the right words to use when expressing feelings during a session. |
| Speed of processing | Peaks in our 20s | A parent driving kids to school can react quickly to unexpected road hazards. |
Dialectical Thinking and Reflective Thought
One form of cognitive processing refined during early adulthood is dialectical thinking , the ability to remain practical, flexible, and open to the multiple factors and solutions in a given situation. Where adolescents think in absolute terms, seeing something as either good or bad, adults are more likely to realize life is full of gray areas and that an event might be a “‘double-edged” sword or a “blessing in disguise”. For example, breaking up with a romantic partner may result in someone finding a more fulfilling relationship or being able to pursue goals they’d previously put on hold. This ability to combine two opposing points of view is at the core of dialectical thinking (Figure 11.18).
Solving adult problems also requires us to constantly consider many sources of information. Reflective thought relies on logical thought processes such as continuous, active evaluation of information and beliefs using evidence and past experiences. It allows us to continually question facts, draw inferences, and make connections between different types of information (Immordino-Yang et al., 2019). It emerges between 20 and 25 years of age, coinciding with brain myelination. Higher education may also stimulate and hone reflective thinking (Sherman, 2021), as students often need to relate new information to old, think through hypothetical situations, and apply their knowledge and skills to solve unfamiliar problems and issues. For example, when learning about the country of Morocco, which is located in Northern Africa near Spain, a learner may be surprised to learn of a Middle Eastern country that is so far from the core group of countries typically associated with the Middle East. In addition, the learner will need and have the ability to hypothesize how Middler Eastern culture may take on different aspects given the distance of Morocco from the Middle East, and its close proximity to Western Europe. Similarly, changes in crystallized intelligence also occur and are believed to be mediated by both an individual’s life experiences and, for those pursuing a degree, the college courses they take (Hochberg & Konner, 2020).
Heuristics
Our thinking also becomes more efficient in early adulthood. Heuristic s are mental shortcuts that help us process information more quickly. They frequently simplify a problem or decision point and reduce the time and energy needed to come up with a solution. Because our brains are metabolically “expensive,” meaning they require a lot of energy to function, being efficient when facing mental tasks makes sense. Common heuristics are the familiarity heuristic, affect heuristic, and scarcity heuristic. All have their advantages and their downsides.
The familiarity heuristic makes us feel more comfortable with things or people we have experience with than with those we have yet to experience or meet. Often, we will choose the most comfortable or familiar option, but then we may never know whether the novel thing or person was a better choice. For example, you may buy apples every week because apples are a familiar choice, not considering less familiar fruits like mangoes, even though you might enjoy mangoes more than apples if you tried them.
The affect heuristic causes us to use emotions more than other factors when making a decision. We can also consider this similar to intuition or a “gut instinct”. Decisions we make using the affect heuristic aren’t always bad—for example, a person on a job interview “gets a bad feeling” about the way the employees interact with each other, declines a job offer, and later learns of a hostile environment lawsuit against that company. However, using the affect heuristic can also cause us to make impractical decisions, such as letting our emotions talk us into adopting a rescue dog even though our living arrangements aren’t appropriate for a dog.
Finally, the scarcity heuristic causes us to believe that the rarer or less available something is, the more valuable it is. Advertisers sometimes count on the scarcity heuristic to sell products, telling us that “quantities are limited” and we should “act now”. Unfortunately, this can cause us to waste money on things we don’t really need.
While heuristics can help us use short-cuts to make decisions, they aren’t always the best way to make a decision. An emerging adult might have ideas about what it would be like to live in a big city based on what they've seen in movies and be excited about all the activities to explore and people to meet, but not realize potential drawbacks, such as the high cost of living, traffic, pollution, and so on. Taking the time to think through a decision like moving to a new area should be done rationally and purposefully, not using a heuristic.
Problem Solving and Divergent Thinking
The use of both convergent and divergent thought to solve problems increases during early adulthood. Convergent thought uses a known solution to deal with a problem, whereas divergent thought creates a new solution. Divergent thought is closely associated with creativity because it often leads to “thinking outside the box” or creating multiple potential ways to solve a problem (Runco, 2014). Although they’re different ways of thinking, convergent and divergent thought can be used together. By combining divergent thought processes with reflective thought, and by honing convergent thought processes, people can gain a deeper understanding of a situation or problem and come up with multiple solutions. A college student who wants to make more friends could use divergent thinking to brainstorm all the possible ways they could do that (e.g., joining clubs, playing intramural sports, attending tutoring sessions) and then use convergent thinking to gather more information (e.g., what available clubs there are) and narrow down options based on factors like time commitment, interests, and other characteristics (e.g., the Chemistry Club is only for chemistry majors, rushing a fraternity would be too expensive).
Social Cognition in Early Adulthood
Social cognition is an area of psychology concerned with what people are thinking within social situations. It asks what information we keep about our previous social interactions, how we use this information in future social interactions, and how we interpret the social motives of others. In this section, we’ll consider several aspects of social cognition: Theory of mind, empathy, moral development, and prosocial behavior.
Theory of Mind and Empathy
Like many other cognitive skills, theory of mind appears in early childhood and continues to develop as we grow up. Cognitive theory of mind is the ability to draw conclusions about another person’s intentions and beliefs, and affective theory of mind is the ability to draw conclusions about another person’s feelings. In general, both types of theory of mind improve as we age (Baksh et al., 2021; Kilford & Blakemore, 2020; Velikonja et al., 2019), giving us a better ability to form connections and interact with others. Empathy also develops in early childhood and improves as we mature (Figure 11.19). As with theory of mind, we can consider both cognitive empathy (being able to adopt someone’s point of view) and affective empathy (being able to vicariously experience someone’s emotions). A potentially important component of empathy is our ability to recognize and process faces, including facial displays of emotion. Several studies have indicated that this ability improves from adolescence to emerging adulthood due to maturation of relevant parts of the brain (Gur & Gur, 2016; Kilford & Blakemore, 2020; Tousignant et al., 2016; Velikonja et al., 2019).
Moral Reasoning and Prosocial Behavior
Another theory that touches on social cognition related to our behavior is Kohlberg’s theory of moral development. Remember from earlier chapters that in Kohlberg’s theory, emerging adults are more likely than teenagers and children to use postconventional reasoning, considering broader ethical principles such as justice and fairness to make moral decisions. However, Kohlberg’s theory has some limitations, such as ignoring the interpersonal aspects of morality and focusing too much on objective justice instead of the needs of others (Miller et al., 2018). Kohlberg (1969) also felt that men advanced farther than in their moral development and viewed women as deficient in their moral reasoning abilities. These ideas met with criticism from, among others, Carol Gilligan , who was a research assistant of Kohlberg. She consequently developed her own ideas of moral development, arguing that women aren’t deficient in their moral reasoning, but rather just reason differently than men do. Specifically, Gilligan proposed that in their moral reasoning, girls and women focus more on staying connected and the importance of interpersonal relationships (Gilligan, 1982). Another criticism of the theory is that Kohlberg’s stories, such as the Heinz dilemma, were culturally biased and did not apply well to all cultures, particularly collectivistic cultures that prioritize interdependence more than independence (Jia & Krettenauer, 2017; Moheghi et al., 2020). Indeed, culture does play a role in cognitive development, as we’ll discover in the next section.
Early adulthood may mean increased moral dilemmas as well as more opportunities to behave in moral ways. Lapsley and colleagues (2023) point out that the moral dilemmas of adult life are often ambiguous and solving them requires considering multiple factors, such as avoiding harm versus doing good, the long-term consequences of the decision, and whether the punishment fits the crime. This maps onto the elements of post-formal thought described earlier. Emerging adults are also faster and more flexible in their moral decision-making than adolescents are (Padilla-Walker, 2016).
Theory of mind, empathy, and moral reasoning can all affect prosocial behavior by giving us more information we can use to help others. When we can both consider other people’s perspectives and identify with them, we can better meet their needs. Being able to consider multiple dimensions of complex moral situations enables us to make decisions that have a better chance of being beneficial. Additionally, in early adulthood we may have more opportunities and more ability to engage in prosocial behavior by, for example, voting or giving a friend a ride somewhere after their car breaks down (Padilla-Walker, 2016).
Link to Learning
How empathic are you? This video gives some interesting perspective on our experience of empathy toward people who are different from us.
Contextual and Cultural Variations in Cognition
People from different cultures approach problems differently, and they value different factors when making decisions. The way an individual forms their identity, and their cultural and societal norms and beliefs, all influence the way they process information and the development of cognitive skills in early adulthood.
Variations in Practical Thinking
In general, collectivistic cultures show a preference for a holistic style of thinking that relies on contextual information, situational attributions, low tolerance for ambiguity, and dislike of competition. Western cultures, on the other hand, tend to be more individualistic and emphasize dispositional (personality) attributions (explanations for behavior), competition, and reliance on absolute facts, with a higher tolerance for ambiguity (Grossman, 2018; Her & Haron, 2016; Rachev & Petkova, 2019). These differences appear to influence some specific aspects of cognition. For example, Rachev and Petkova discovered that Bulgarian business owners were extremely susceptible to the affect heuristic, and attributed this to a collectivistic tendency to prioritize emotional satisfaction over competition; in other words, because emotional factors are so important, they become a primary focus when making decisions (2019). A study comparing Israeli and Taiwanese children ages 4–11 found that the Israeli children demonstrated the scarcity heuristic by age 7, while the Taiwanese children never showed it. The researchers explained this difference in individualistic/collectivistic terms: When an item is viewed as scarce, members of individualistic cultures are more likely to want it because it helps them stand out, but this isn’t a priority for members of individualistic cultures (Diesendruck et al., 2018). While this is child research and not adult research, it provides more support for the idea that cultural factors may influence use of specific cognitive strategies, and that this influence may start well before adulthood.
Culture may also affect views of convergent and divergent thinking as well as creativity. Specifically, collectivistic cultures may prioritize convergent thinking over divergent thinking because it focuses on finding the “right answer” and therefore reducing uncertainty (Her & Haron, 2016). Consistent with this, Paletz and colleagues note that Eastern and Western cultures tend to define creativity somewhat differently; Eastern cultures tend to define creativity in terms of appropriateness to the context of prevailing cultural norms, while Western cultures tend to define it in terms of novelty (2018). This may influence the way people generate solutions to problems. For example, a person from a Western culture may consider all possible solutions to a problem, regardless of how feasible they are, but someone from an Eastern culture may only consider solutions that are practical (Figure 11.20).
Life circumstances other than culture can also affect adult cognition. Research comparing the use of availability heuristics in Italy and Sweden has demonstrated that while residents of both countries use these heuristics, the focus varies depending on what’s most salient to residents of those countries. Specifically, when asked to estimate the likelihood of flood, drought, and wildfires, the respondents rated as more likely the specific events that had occurred in their country most recently (Di Baldassarre et al., 2021). Thus, while the ability to use heuristics is universal, the information that we use to shape our judgments depends on experience.
Variations in Social Cognition
Specific cultural values may affect social cognition. For example, the value of familism , prevalent in Latino culture, emphasizes the importance of helping others and is related to expressed prosocial tendencies (Knight et al., 2016; 2018). A study of college students in India, Japan, and the United States found that the Japanese and American students were more likely to use exchange norm s (either repaying a debt or anticipating future benefit) when deciding whether to help a friend, while Indian students were more likely to use communal norm s (helping with no expectation of reciprocity; Miller et al., 2017). Also, a study comparing European American and Hindu Indian adults found that when evaluating the reasons why a person did something (e.g., didn’t help someone in need), European American participants tended to attribute this behavior to personality characteristics (e.g., being selfish), while Hindu Indian participants tended to attribute it to contextual factors (e.g., not wanting to be late for work; Miller et al., 2018). This suggests that while the ability to consider others’ inner states is common across cultures, there may be cultural differences in what people prioritize when making those judgments Figure 11.21.
Contextual factors other than culture may also be relevant. For example, people who experience autism spectrum disorder have more difficulty than neurotypical people on tasks requiring theory of mind (e.g., Baksh et al., 2021; Sasson et al., 2019), although this may depend to some extent on underlying general intellectual ability (Sasson et al., 2019). Also, when asked to explain their motivation for performing prosocial acts, people with autism spectrum disorder tend to provide egocentric answers (“I’m nice”) instead of answers that indicate larger understanding of social norms (“It’s important to help people”; Baksh et al., 2021).
References
Baksh, R. A., Abrahams, S., Bertlich, M., Cameron, R., Jany, S., Dorrian, T., Baron-Cohen, S., Allison, C., Smith, P., MacPherson, S. E., & Auyeung, B. (2021). Social cognition in adults with autism spectrum disorders: Validation of the Edinburgh Social Cognition Test (ESCoT). The Clinical Neuropsychologist, 35 (7), 1275-1293. doi.org/10.1080/13854046.2020.1737236
Brodbeck, J., Bachmann, M. S., Croudace, T. J., & Brown, A. (2013). Comparing growth trajectories of risk behaviors from late adolescence through young adulthood: An accelerated design. Developmental Psychology, 49 (9), 1732–1738. doi.org/10.1037/a0030873
Cohen, A. O., Breiner, K., Steinberg, L., Bonnie, R. J., Scott, E. S., Taylor-Thompson, K., Rudolph, M. D., Chein, J., Richeson, J. A., Heller, A. S., Silverman, M. R., Dellarco, D. V., Fair, D. A., Galván, A., & Casey, B. J. (2016). When is an adolescent an adult? Assessing cognitive control in emotional and nonemotional contexts. Psychological Science , 27 (4), 549–562. doi.org/10.1177/0956797615627625
Dennis, E. L., Jahanshad, N., McMahon, K. L., de Zubicaray, G. I., Martin, N. G., Hickie, I. B., Toga, A. W., Wright, M. J., & Thompson, P. M. (2013). Development of brain structural connectivity between ages 12 and 30: A 4-Tesla diffusion imaging study in 439 adolescents and adults. NeuroImage , 64 , 671–684. https://doi.org/10.1016/j.neuroimage.2012.09.004
Di Baldassarre, G., Mondino, E., Rusca, M., Del Giudice, E., Mård, J., Ridolfi, E., Scolobig, A., & Raffetti, E. (2021). Multiple hazards and risk perceptions over time: the availability heuristic in Italy and Sweden under COVID-19. Natural Hazards and Earth System Sciences Discussions, 24 (11), 3439-3447. https://doi.org/10.5194/nhess-21-3439-2021
Diesendruck, G., Chiang, W. C., Ferera, M., & Benozio, A. (2019). Cultural differences in the development of a preference for scarce objects. Developmental Psychology, 55 (1), 89. doi.org/10.1037/dev0000620
Dvorak, A. V., Swift-LaPointe, T., Vavasour, I. M., Lee, L. E., Abel, S., Russell-Schulz, B., Graf, C., Wurl, A., Liu, H., Laule, C., Li, D. K. B., Traboulsee, A., Tam, R., Boyd, L. A., MacKay, A. L., & Kolind, S. H. (2021). An atlas for human brain myelin content throughout the adult life span. Scientific Reports, 11 (1), 269. https://doi.org/10.1038/s41598-020-79540-3
Ferguson, H. J., Brunsdon, V. E. A., & Bradford, E. E. F. (2021). The developmental trajectories of executive function from adolescence to old age. Scientific Reports, 11 , 1382. https://doi.org/10.1038/s41598-020-80866-1
Gilligan, C. (1982). In a different voice: Psychological theory and women's development . Harvard University Press.
Griffin, J., Gooding, S., Semesky, M., Farmer, B., Mannchen, G., & Sinnott, J. (2009). Four brief studies of relations between postformal thought and non-cognitive factors: Personality, concepts of God, political opinions, and social attitudes. J ournal of Adult Development, 16 (3), 173–182. https://doi.org/10.1007/s10804-009-9056-0
Grossmann, I. (2018). Dialecticism across the lifespan: Toward a deeper understanding of the ontogenetic and cultural factors influencing dialectical thinking and emotional experience. In J. Spencer-Rodgers & K. Peng (Eds.), The psychological and cultural foundations of East Asian cognition: Contradiction, change, and holism (pp. 135–180). Oxford University Press.
Gur, R. C., & Gur, R. E. (2016). Social cognition as an RDoC domain. American Journal of Medical Genetics. Part B, Neuropsychiatric Genetics, 171B (1), 132–141. doi.org/10.1002/ajmg.b.32394
Hartshorne, J. K., & Germine, L. T. (2015). When does cognitive functioning peak? The asynchronous rise and fall of different cognitive abilities across the life span. Psychological Science, 26 (4), 433-443. doi.org/10.1177/0956797614567339
Her, E. L. W., & Haron, F. (2016). The association of giftedness, creativity, and postformal thinking in Malaysian adults. Advanced Development, 15 , 47-62. www.proquest.com/scholarly-j...781544923/se-2
Hochberg, Z., & Konner, M. (2020). Emerging adulthood, a pre-adult life- history stage. Frontiers in Endocrinology, 10 (918). https://doi.org/10.3389/fendo.2019.00918
Immordino-Yang, M. H., Darling-Hammond, L., & Krone, C. R. (2019). Nurturing nature: How brain development is inherently social and emotional, and what this means for education. Educational Psychologist, 54 (3), 185–204. doi.org/10.1080/00461520.2019.1633924
Jia, F., & Krettenauer, T. (2017). Recognizing moral identity as a cultural construct. Frontiers in Psychology, 8 , 412. https://doi.org/10.3389/fpsyg.2017.00412
Kilford, E. J., Garrett, E., & Blakemore, S. J. (2016). The development of social cognition in adolescence: An integrated perspective. Neuroscience and Biobehavioral Reviews, 70 , 106–120. https://doi.org/10.1016/j.neubiorev.2016.08.016
Kincheloe, J. L., & Steinberg, S. R. (1993). A tentative description of post-formal thinking: The critical confrontation with cognitive theory. Harvard Educational Review, 63 (3), 296-320. www.proquest.com/scholarly-j...212279037/se-2
Knight, G. P., Carlo, G., Mahrer, N. E., & Davis, A. N. (2016). The socialization of culturally related values and prosocial tendencies among Mexican American Adolescents. Child Development, 87 (6), 1758-1771. doi.org/10.1111/cdev.12634
Knight, G. P., Mazza, G. L., & Carlo, G. (2018). Trajectories of familism values and the prosocial tendencies of Mexican American adolescents. Developmental Psychology, 54 (2), 378-384. doi.org/10.1037/dev0000436
Kohlberg, L. (1969). Stage and sequence: The cognitive-developmental approach to socialization. In D. A. Goslin (Ed.), Handbook of socialization theory (pp. 347-480). Rand McNally.
Kwon, D., Pfefferbaum, A., Sullivan, E. V., & Pohl, K. M. (2020). Regional growth trajectories of cortical myelination in adolescents and young adults: longitudinal validation and functional correlates. Brain Imaging and Behavior, 14 (1), 242–266. https://doi.org/10.1007/s11682-018-9980-3
Lapsley, D., LaPorte, E., & Kelley, K. (2023). Moral cognition in adolescence and emerging adulthood. In L. J. Crockett, G. Carlo, & J. E. Schulenberg (Eds.), APA handbook of adolescent and young adult development (pp. 123–138). American Psychological Association. doi.org/10.1037/0000298-008
Lee, J., & Kim, H. J. (2022). Normal aging induces changes in the brain and neurodegeneration progress: review of the structural, biochemical, metabolic, cellular, and molecular changes. Frontiers in Aging Neuroscience, 14 , 931536. https://doi.org/10.3389/fnagi.2022.931536
Miller, J. G., Akiyama, H., & Kapadia, S. (2017). Cultural variation in communal versus exchange norms: Implications for social support. Journal of Personality and Social Psychology, 113 (1), 81–94. doi.org/10.1037/pspi0000091
Miller, J. G., Wice, M., & Goyal, N. (2018). Contributions and challenges of cultural research on the development of social cognition. Developmental Review, 50 (Part A), 65–76. https://doi.org/10.1016/j.dr.2018.03.003
Mills, K. L., Siegmund, K. D., Tamnes, C. K., Ferschmann, L., Wierenga, L. M., Bos, M. G. N., Luna, B., Li, C., & Herting, M. M. (2021). Inter-individual variability in structural brain development from late childhood to young adulthood. NeuroImage, 242 , 118450. https://doi.org/10.1016/j.neuroimage.2021.118450
Moheghi, M., Ghorbanzadeh, M., & Abedi, J. (2020). The investigation and criticism moral development ideas of Kohlberg, Piaget and Gilligan. International Journal of Multicultural and Multireligious Understanding, 7 (2), 362-374. http://dx.doi.org/10.18415/ijmmu.v7i2.1516
Padilla-Walker, L. M. (2016). Moral development during emerging adulthood. In J. J. Arnett (Ed.), The Oxford handbook of emerging adulthood (pp. 449–463). Oxford University Press.
Paletz, S. B. F., Bogue, K., Miron-Spektor, E., & Spencer-Rodgers, J. (2018). Dialectical thinking and creativity from many perspectives: Contradiction and tension. In J. Spencer-Rodgers & K. Peng (Eds.), The psychological and cultural foundations of East Asian cognition: Contradiction, change, and holism (pp. 267–308). Oxford University Press.
Parr, A. C., Calabro, F., Tervo-Clemmens, B., Larsen, B., Foran, W., & Luna, B. (2022). Contributions of dopamine-related basal ganglia neurophysiology to the developmental effects of incentives on inhibitory control. Developmental Cognitive Neuroscience, 54 , 101100. https://doi.org/10.1016/j.dcn.2022.101100
Rachev, N. R., & Petkova, M. (2019). How prone are Bulgarians to heuristics and biases? Implications for studying rationality across cultures. Journal of Cognition and Culture, 19 (3-4), 322-342. https://doi.org/10.1163/15685373-12340062
Runco, M. A. (2014). Creativity theories and themes: Research, development, and practice. Academic Press. https://doi.org/10.1016/B978-0-12-410512-6.00001-1
Sasson, N. J., Morrison, K. E., Kelsven, S., & Pinkham, A. E. (2020). Social cognition as a predictor of functional and social skills in autistic adults without intellectual disability. Autism Research, 13 (2), 259-270. doi.org/10.1002/aur.2195
Schreiber, L. R. N., Grant, J. E., & Odlaug, B. L. (2012). Emotion regulation and impulsivity in young adults. Journal of Psychiatric Research , 46 (5), 651–658. https://doi.org/10.1016/j.jpsychires.2012.02.005
Sherman, G. L. (2021). Transformative learning and well-being for emerging adults in higher education. Journal of Transformative Education, 19 (1), 29-49. doi.org/10.1177/1541344620935623
Simpson, A. R. (2018). Brain changes . Young Adult Development Project. https://hr.mit.edu/static/worklife/y...ult/brain.html
Sinnott, J. D. (1998). What characterizes postformal thought? In The development of logic in adulthood (pp. 23-49). Springer. https://doi.org/10.1007/978-1-4757-2911-5_3
Stebbins, R. C., Yang, Y. C., Reason, M., Aiello, A. E., Belsky, D. W., Harris, K. M., & Plassman, B. L. (2022). Occupational cognitive stimulation, socioeconomic status, and cognitive functioning in young adulthood. SSM-Population Health, 17 , 101024. https://doi.org/10.1016/j.ssmph.2022.101024
Taber-Thomas, B., & Pérez-Edgar, K. (2015). Emerging adulthood brain development. In J. J. Arnett (Ed.), The Oxford handbook of emerging adulthood (pp. 126-141). Oxford University Press. doi.org/10.1093/oxfordhb/9780199795574.013.15
Tousignant, B., Sirois, K., Achim, A. M., Massicotte, E., & Jackson, P. L. (2017). A comprehensive assessment of social cognition from adolescence to adulthood. Cognitive Development, 43 , 214–223. https://doi.org/10.1016/j.cogdev.2017.05.001
U.S. Department of Transportation, Bureau of Transportation Statistics (2023). Transportation statistics annual report 2023 . https://doi.org/10.21949/1529944
Velikonja, T., Fett, A. K., & Velthorst, E. (2019). Patterns of nonsocial and social cognitive functioning in adults with autism spectrum disorder: A systematic review and meta-analysis. JAMA Psychiatry, 76 (2), 135–151. doi.org/10.1001/jamapsychiatry.2018.3645
Whitley, E., Deary, I. J., Ritchie, S. J., Batty, G. D., Kumari, M., & Benzeval, M. (2016). Variations in cognitive abilities across the life course: Cross-sectional evidence from Understanding Society: The UK Household Longitudinal Study. Intelligence, 59 , 39-50. https://doi.org/10.1016/j.intell.2016.07.001
Wood, D., Crapnell, T., Lau, L., Bennett, A., Lotstein, D., Ferris, M., & Kuo, A. (2017). Emerging adulthood as a critical stage in the life course. In N. Halfon, C. B. Forrest, Lerner, R. M., & E. M. Faustman (Eds.), Handbook of life course health development (pp. 123–143). Springer. https://library.oapen.org/bitstream/...7.pdf#page=138


2.20: Slideshow- The Late Baroque
You have probably already noticed that all the pieces on the playlist for this module sound quite different from one another. Setting aside differences of instrumentation and genre, there are significant differences between pieces written in the early, middle, and late portions of the Baroque period. As has been already mentioned, this is a period of considerable musical evolution.
Some of the hallmarks of late Baroque style are a steady, driving beat, lengthy melodic lines (often extended through the use of repeated ascending or descending melodic patterns known as sequences), and dense textures. The composers in the early Baroque were trying to get away from the complexity of late Renaissance polyphonic texture. As a result, they strove for simple homophonic textures. By the late Baroque, composers had once again embraced the complexity of elaborate polyphony. This is especially true of German composers writing for German audiences. Johann Sebastian Bach is a perfect example of this Germanic preference in the late Baroque.
Of course, neither Bach nor Handel were limited by regional tastes. Handel received his musical education in both Germany and Italy, and Bach studied the music of Italian greats Corelli and Vivaldi during his employment as music director at the ducal court in Weimar between 1708 and 1717. Both composers were able to combine their mastery of counterpoint with the melodic lyricism and lighter textures of the Italian style. This combination of exquisite contrapuntal technique and international influences, along with their astonishing talent, is the main reason we view these two composers and their music as representing the culmination of a century and a half of Baroque music.
This presentation covers the late Baroque music of Bach and Handel. Please use this as a study guide during and after your online reading.
Contributors and Attributions
- Authored by : Elliott Jones. Provided by : Santa Ana College. Located at : http://www.sac.edu . License : CC BY: Attribution


2.1: Lab 2- Bones and Bone Markings
Measurable Outcomes
- Determine if a given bone is part of the axial or appendicular skeleton.
- Ascertain the major bones of the skull, as well as any markings or unique features, the regions of the vertebral column, parts of a typical vertebra, along with the other bones and features of the axial skeleton.
- Identify the bones of the appendicular skeleton and their unique features.
- Designate bones as either left or right when applicable. Examples include the ulna, humerus, femurs, scapulas, and clavicles.
- Understand how different bones fit together and articulate. Demonstrate this by assembling different regions of the body using the bones provided.
- Differentiate compact, spongy and dry bone histology slides. This includes identifying the unique characteristics of each.
- Demonstrate an adequate understand of the material in this section.
Background
The skeletal system is the primary structural organ system of the body. Many people think of the skeletal system as being static in that it is unchanging, however, this is not the case. Bones, like other organ systems, have specialized cells which allow them to perform a variety of essential tasks. Osteoblast are responsible for secreting the bony matrix necessary for bone formation. Osteoclast, meanwhile, are large multinucleated cells responsible for the dissolution and reabsorption of bone. It is made mostly of collagen, which gives bone its soft framework, and calcium phosphate which adds strength and hardness to the structure. It is divided into the axial and the appendicular skeleton. The axial skeleton consists of the skull, hyoid bone, vertebral column, sternum, and ribs. Whereas the appendicular skeleton consists of the clavicle, scapula and the rest of the upper and lower limbs. Without the foundational structure of the skeletal system, there would be nothing to support the body and provide points of attachment for muscles. Bones function to protect internal organs, assist body movements, store and release calcium and phosphorous, participate in blood cell production and store fat in the yellow marrow. Bones also function to protect internal organs, assist body movements, and the storage and release of ions such as calcium and phosphorous. Furthermore, long bones contain both hemopoietic (red) and stromal (yellow) marrow which produce red blood cells and fat cells respectively. Each of these cells have specific functions that are key to the development and repair of a bone over time. The two types of bone tissue are compact and spongy bone. Compact bone is typically found along the perimeter of bones and makes up the majority of the diaphysis of long bones. It is stronger than spongy bone and provides more stability. Compact bone is made up of circular units called osteons. Osteons are composed of rings called lamellae that spiral down into a central canal, known as the Haversian canal . This central canal is the passage for nerves, blood vessels, and lymphatics. Spongy bone , on the other hand, is typically the deepest layer of a bone’s composition. It is made of trabeculae which give spongy bone its characteristic lighter weight. There are five classifications of bones based on their shape, long bones, short bones, flat bones, irregular bones and sesamoid bones. The shape and composition of each bone allow them to function as mentioned above.
Vocabulary for Bones and Bone Markings on page(s) 161-162.


3.8: MATERIAL CONNOTATIONS OF CLASS OR STATION
Other more mundane materials and appropriated components might also have strong political connotations that intensify the meaning of the artwork. Korean artist Do Ho Su chose and assembled military dog tags to create a larger-than-life figural impression of an imperialistic robe with a hollow core. It carries connotations of the political strength of his native land being built upon such things as the dehumanizing mandatory military service he had performed, and the relationships between individuals and the collectives they form. ( Some/One , Do Ho Suh:2yhr3j6imaw4e4zzg38k38ar-wpe...loads/2016/10/ suh-inst-002.jpg; Some/One detail, Do Ho Suh: 2yhr3j6imaw4e4zzg38k38ar-wpengine. netdna-ssl.com/wp-content/uploads/2016/10/suh-inst-001.540.jpg)


7.5: Software Simulation Tools
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There are numerous tools designed specifically for the simulation of computer screens by recording screen interactions. For example:
Many of these also do PowerPoint to Flash conversion.
Most course authoring tools can create and deliver tests and quizzes as part of the courses. Instructors may also want use test banks that publishers provide, and/or other, more powerful tools built specifically for testing. For example:
There are well over 100 available sources for software that can be categorized as course authoring tools. When choosing an LMS, be sure that it can support any third-party generic courseware or content authoring tools being used. Particular attention should be paid to the LMS’s ability to launch the courses, and track and record interactions and responses to quizzes. Support for standards helps, but it is no guarantee. You should test the LMS with the tools and courseware that you will be using. You should also determine how accessible the file formats are for students with disabilities. (See Chapter 11, Accessibility and Universal Design, for more information about accessibility.)
Tip
Be careful with rapid development tools. Speed of delivery can be very important but make sure you are not just making bad Word or PowerPoint documentation into even worse e-learning courses.


3.4b: Double Integrals in Polar Coordinates
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Learning Objectives
- Recognize the format of a double integral over a polar rectangular region.
- Evaluate a double integral in polar coordinates by using an iterated integral.
- Recognize the format of a double integral over a general polar region.
- Use double integrals in polar coordinates to calculate areas and volumes.
Double integrals are sometimes much easier to evaluate if we change rectangular coordinates to polar coordinates. However, before we describe how to make this change, we need to establish the concept of a double integral in a polar rectangular region.
Polar Rectangular Regions of Integration
When we defined the double integral for a continuous function in rectangular coordinates—say, \(g\) over a region \(R\) in the \(xy\)-plane—we divided \(R\) into subrectangles with sides parallel to the coordinate axes. These sides have either constant \(x\)-values and/or constant \(y\)-values. In polar coordinates, the shape we work with is a polar rectangle , whose sides have constant \(r\)-values and/or constant \(\theta\)-values. This means we can describe a polar rectangle as in Figure \(\PageIndex{1a}\), with \(R = \{(r,\theta)\,|\, a \leq r \leq b, \, \alpha \leq \theta \leq \beta\}\).
In this section, we are looking to integrate over polar rectangles. Consider a function \(f(r,\theta)\) over a polar rectangle \(R\). We divide the interval \([a,b]\) into \(m\) subintervals \([r_{i-1}, r_i]\) of length \(\Delta r = (b - a)/m\) and divide the interval \([\alpha, \beta]\) into \(n\) subintervals \([\theta_{i-1}, \theta_i]\) of width \(\Delta \theta = (\beta - \alpha)/n\). This means that the circles \(r = r_i\) and rays \(\theta = \theta_i\) for \(1 \leq i \leq m\) and \(1 \leq j \leq n\) divide the polar rectangle \(R\) into smaller polar subrectangles \(R_{ij}\) (Figure \(\PageIndex{1b}\)).
As before, we need to find the area \(\Delta A\) of the polar subrectangle \(R_{ij}\) and the “polar” volume of the thin box above \(R_{ij}\). Recall that, in a circle of radius \(r\) the length \(s\) of an arc subtended by a central angle of \(\theta\) radians is \(s = r\theta\). Notice that the polar rectangle \(R_{ij}\) looks a lot like a trapezoid with parallel sides \(r_{i-1}\Delta \theta\) and \(r_i\Delta \theta\) and with a width \(\Delta r\). Hence the area of the polar subrectangle \(R_{ij}\) is
\[\Delta A = \frac{1}{2} \Delta r (r_{i-1} \Delta \theta + r_i \Delta \theta ). \nonumber \]
Simplifying and letting
\[r_{ij}^* = \frac{1}{2}(r_{i-1}+r_i) \nonumber \]
we have \(\Delta A = r_{ij}^* \Delta r \Delta \theta\).
Therefore, the polar volume of the thin box above \(R_{ij}\) (Figure \(\PageIndex{2}\)) is
\[f(r_{ij}^*, \theta_{ij}^*) \Delta A = f(r_{ij}^*, \theta_{ij}^*)r_{ij}^* \Delta r \Delta \theta. \nonumber \]
Using the same idea for all the subrectangles and summing the volumes of the rectangular boxes, we obtain a double Riemann sum as
\[\sum_{i=1}^m \sum_{j=1}^n f(r_{ij}^*, \theta_{ij}^*) r_{ij}^* \Delta r \Delta \theta. \nonumber \]
As we have seen before, we obtain a better approximation to the polar volume of the solid above the region \(R\) when we let \(m\) and \(n\) become larger. Hence, we define the polar volume as the limit of the double Riemann sum,
\[V = \lim_{m,n\rightarrow\infty}\sum_{i=1}^m \sum_{j=1}^n f(r_{ij}^*, \theta_{ij}^*) r_{ij}^* \Delta r \Delta \theta. \nonumber \]
This becomes the expression for the double integral.
Definition: The double integral in polar coordinates
The double integral of the function \(f(r, \theta)\) over the polar rectangular region \(R\) in the \(r\theta\)-plane is defined as
\[\begin{align} \iint_R f(r, \theta)dA &= \lim_{m,n\rightarrow\infty}\sum_{i=1}^m \sum_{j=1}^n f(r_{ij}^*, \theta_{ij}^*) \Delta A \\[4pt] &= \lim_{m,n\rightarrow \infty} \sum_{i=1}^m \sum_{j=1}^n f(r_{ij}^*,\theta_{ij}^*)r_{ij}^* \Delta r \Delta \theta. \end{align} \nonumber \]
Again, just as in section on Double Integrals over Rectangular Regions, the double integral over a polar rectangular region can be expressed as an iterated integral in polar coordinates. Hence,
\[\iint_R f(r, \theta)\,dA = \iint_R f(r, \theta) \,r \, dr \, d\theta = \int_{\theta=\alpha}^{\theta=\beta} \int_{r=a}^{r=b} f(r,\theta) \,r \, dr \, d\theta. \nonumber \]
Notice that the expression for \(dA\) is replaced by \(r \, dr \, d\theta\) when working in polar coordinates. Another way to look at the polar double integral is to change the double integral in rectangular coordinates by substitution. When the function \(f\) is given in terms of \(x\) and \(y\) using \(x = r \, \cos \, \theta, \, y = r \, \sin \, \theta\), and \(dA = r \, dr \, d\theta\) changes it to
\[\iint_R f(x,y) \,dA = \iint_R f(r \, \cos \, \theta, \, r \, \sin \, \theta ) \,r \, dr \, d\theta. \nonumber \]
Note that all the properties listed in section on Double Integrals over Rectangular Regions for the double integral in rectangular coordinates hold true for the double integral in polar coordinates as well, so we can use them without hesitation.
Example \(\PageIndex{1A}\): Sketching a Polar Rectangular Region
Sketch the polar rectangular region
\[R = \{(r, \theta)\,|\,1 \leq r \leq 3, 0 \leq \theta \leq \pi \}. \nonumber \]
Solution
As we can see from Figure \(\PageIndex{3}\), \(r = 1\) and \(r = 3\) are circles of radius 1 and 3 and \(0 \leq \theta \leq \pi\) covers the entire top half of the plane. Hence the region \(R\) looks like a semicircular band.
Now that we have sketched a polar rectangular region, let us demonstrate how to evaluate a double integral over this region by using polar coordinates.
Example \(\PageIndex{1B}\): Evaluating a Double Integral over a Polar Rectangular Region
Evaluate the integral \(\displaystyle \iint_R 3x \, dA\) over the region \(R = \{(r, \theta)\,|\,1 \leq r \leq 2, \, 0 \leq \theta \leq \pi \}.\)
Solution
First we sketch a figure similar to Figure \(\PageIndex{3}\), but with outer radius \(r=2\). From the figure we can see that we have
\[\begin{align*} \iint_R 3x \, dA &= \int_{\theta=0}^{\theta=\pi} \int_{r=1}^{r=2} 3r \, \cos \, \theta \,r \, dr \, d\theta \quad\text{Use an integral with correct limits of integration.} \\ &= \int_{\theta=0}^{\theta=\pi} \cos \, \theta \left[\left. r^3\right|_{r=1}^{r=2}\right] d\theta \quad\text{Integrate first with respect to $r$.} \\ &=\int_{\theta=0}^{\theta=\pi} 7 \, \cos \, \theta \, d\theta \\ &= 7 \, \sin \, \theta \bigg|_{\theta=0}^{\theta=\pi} = 0. \end{align*}\]
Exercise \(\PageIndex{1}\)
Sketch the region \(D = \{ (r,\theta) \vert 1\leq r \leq 2, \, -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2} \}\), and evaluate \(\displaystyle \iint_R x \, dA\).
- Hint
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Follow the steps in Example \(\PageIndex{1A}\).
- Answer
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\(\frac{14}{3}\)
Example \(\PageIndex{2A}\): Evaluating a Double Integral by Converting from Rectangular Coordinates
Evaluate the integral
\[\iint_R (1 - x^2 - y^2) \,dA \nonumber \]
where \(R\) is the unit circle on the \(xy\)-plane.
Solution
The region \(R\) is a unit circle, so we can describe it as \(R = \{(r, \theta )\,|\,0 \leq r \leq 1, \, 0 \leq \theta \leq 2\pi \}\).
Using the conversion \(x = r \, \cos \, \theta, \, y = r \, \sin \, \theta\), and \(dA = r \, dr \, d\theta\), we have
\[\begin{align*} \iint_R (1 - x^2 - y^2) \,dA &= \int_0^{2\pi} \int_0^1 (1 - r^2) \,r \, dr \, d\theta \\[4pt] &= \int_0^{2\pi} \int_0^1 (r - r^3) \,dr \, d\theta \\ &= \int_0^{2\pi} \left[\frac{r^2}{2} - \frac{r^4}{4}\right]_0^1 \,d\theta \\&= \int_0^{2\pi} \frac{1}{4}\,d\theta = \frac{\pi}{2}. \end{align*}\]
Example \(\PageIndex{2B}\): Evaluating a Double Integral by Converting from Rectangular Coordinates
Evaluate the integral \[\displaystyle \iint_R (x + y) \,dA \nonumber \] where \(R = \big\{(x,y)\,|\,1 \leq x^2 + y^2 \leq 4, \, x \leq 0 \big\}.\)
Solution
We can see that \(R\) is an annular region that can be converted to polar coordinates and described as \(R = \left\{(r, \theta)\,|\,1 \leq r \leq 2, \, \frac{\pi}{2} \leq \theta \leq \frac{3\pi}{2} \right\}\) (see the following graph).
Hence, using the conversion \(x = r \, \cos \, \theta, \, y = r \, \sin \, \theta\), and \(dA = r \, dr \, d\theta\), we have
\[\begin{align*} \iint_R (x + y)\,dA &= \int_{\theta=\pi/2}^{\theta=3\pi/2} \int_{r=1}^{r=2} (r \, \cos \, \theta + r \, \sin \, \theta) r \, dr \, d\theta \\ &= \left(\int_{r=1}^{r=2} r^2 \, dr\right)\left(\int_{\pi/2}^{3\pi/2} (\cos \, \theta + \sin \, \theta)\,d\theta\right) \\ &= \left. \left[\frac{r^3}{3}\right]_1^2 [\sin \, \theta - \cos \, \theta] \right|_{\pi/2}^{3\pi/2} \\ &= - \frac{14}{3}. \end{align*}\]
Exercise \(\PageIndex{2}\)
Evaluate the integral \[ \displaystyle \iint_R (4 - x^2 - y^2)\,dA \nonumber \] where \(R\) is the circle of radius 2 on the \(xy\)-plane.
- Hint
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Follow the steps in the previous example.
- Answer
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\(8\pi\)
General Polar Regions of Integration
To evaluate the double integral of a continuous function by iterated integrals over general polar regions, we consider two types of regions, analogous to Type I and Type II as discussed for rectangular coordinates in section on Double Integrals over General Regions. It is more common to write polar equations as \(r = f(\theta)\) than \(\theta = f(r)\), so we describe a general polar region as \(R = \{(r, \theta)\,|\,\alpha \leq \theta \leq \beta, \, h_1 (\theta) \leq r \leq h_2(\theta)\}\) (Figure \(\PageIndex{5}\)).
Theorem: Double Integrals over General Polar Regions
If \(f(r, \theta)\) is continuous on a general polar region \(D\) as described above, then
\[\iint_D f(r, \theta ) \,r \, dr \, d\theta = \int_{\theta=\alpha}^{\theta=\beta} \int_{r=h_1(\theta)}^{r=h_2(\theta)} f(r,\theta) \, r \, dr \, d\theta. \nonumber \]
Example \(\PageIndex{3}\): Evaluating a Double Integral over a General Polar Region
Evaluate the integral
\[\iint_D r^2 \sin \theta \, r \, dr \, d\theta \nonumber \]
where \(D\) is the region bounded by the polar axis and the upper half of the cardioid \(r = 1 + \cos \, \theta\).
Solution
We can describe the region \(D\) as \(\{(r, \theta)\,|\,0 \leq \theta \leq \pi, \, 0 \leq r \leq 1 + \cos \, \theta\} \) as shown in Figure \(\PageIndex{6}\).
Hence, we have
\[\begin{align*} \iint_D r^2 \sin \, \theta \, r \, dr \, d\theta &= \int_{\theta=0}^{\theta=\pi} \int_{r=0}^{r=1+\cos \theta} (r^2 \sin \, \theta) \,r \, dr \, d\theta \\ &= \frac{1}{4}\left.\int_{\theta=0}^{\theta=\pi}[r^4] \right|_{r=0}^{r=1+\cos \, \theta} \sin \, \theta \, d\theta \\ &= \frac{1}{4} \int_{\theta=0}^{\theta=\pi} (1 + \cos \, \theta )^4 \sin \, \theta \, d\theta \\ &= - \frac{1}{4} \left[ \frac{(1 + \cos \, \theta)^5}{5}\right]_0^{\pi} = \frac{8}{5}.\end{align*}\]
Exercise \(\PageIndex{3}\)
Evaluate the integral
\[\iint_D r^2 \sin^2 2\theta \,r \, dr \, d\theta \nonumber \]
where \(D = \left\{ (r,\theta)\,|\,0 \leq \theta \leq \pi, \, 0 \leq r \leq 2 \sqrt{\cos \, 2\theta} \right\}\).
- Hint
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Graph the region and follow the steps in the previous example.
- Answer
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\(\frac{\pi}{8}\)
Polar Areas and Volumes
As in rectangular coordinates, if a solid \(S\) is bounded by the surface \(z = f(r, \theta)\), as well as by the surfaces \(r = a, \, r = b, \, \theta = \alpha\), and \(\theta = \beta\), we can find the volume \(V\) of \(S\) by double integration, as
\[V = \iint_R f(r, \theta) \,r \, dr \, d\theta = \int_{\theta=\alpha}^{\theta=\beta} \int_{r=a}^{r=b} f(r,\theta)\, r \, dr \, d\theta. \nonumber \]
If the base of the solid can be described as \(D = \{(r, \theta)|\alpha \leq \theta \leq \beta, \, h_1 (\theta) \leq r \leq h_2(\theta)\}\), then the double integral for the volume becomes
\[V = \iint_D f(r, \theta) \,r \, dr \, d\theta = \int_{\theta=\alpha}^{\theta=\beta} \int_{r=h_1(\theta)}^{r=h_2(\theta)} f(r,\theta) \,r \, dr \, d\theta. \nonumber \]
We illustrate this idea with some examples.
Example \(\PageIndex{4A}\): Finding a Volume Using a Double Integral
Find the volume of the solid that lies under the paraboloid \(z = 1 - x^2 - y^2\) and above the unit circle on the \(xy\)-plane (Figure \(\PageIndex{7}\)).
Solution
By the method of double integration, we can see that the volume is the iterated integral of the form
\[\displaystyle \iint_R (1 - x^2 - y^2)\,dA \nonumber \]
where \(R = \big\{(r, \theta)\,|\,0 \leq r \leq 1, \, 0 \leq \theta \leq 2\pi\big\}\).
This integration was shown before in Example \(\PageIndex{2A}\), so the volume is \(\frac{\pi}{2}\) cubic units.
Example \(\PageIndex{4B}\): Finding a Volume Using Double Integration
Find the volume of the solid that lies under the paraboloid \(z = 4 - x^2 - y^2\) and above the disk \((x - 1)^2 + y^2 = 1\) on the \(xy\)-plane. See the paraboloid in Figure \(\PageIndex{8}\) intersecting the cylinder \((x - 1)^2 + y^2 = 1\) above the \(xy\)-plane.
Solution
First change the disk \((x - 1)^2 + y^2 = 1\) to polar coordinates. Expanding the square term, we have \(x^2 - 2x + 1 + y^2 = 1\). Then simplify to get \(x^2 + y^2 = 2x\), which in polar coordinates becomes \(r^2 = 2r \, \cos \, \theta\) and then either \(r = 0\) or \(r = 2 \, \cos \, \theta\). Similarly, the equation of the paraboloid changes to \(z = 4 - r^2\). Therefore we can describe the disk \((x - 1)^2 + y^2 = 1\) on the \(xy\) -plane as the region
\[D = \{(r,\theta)\,|\,0 \leq \theta \leq \pi, \, 0 \leq r \leq 2 \, \cos \theta\}. \nonumber \]
Hence the volume of the solid bounded above by the paraboloid \(z = 4 - x^2 - y^2\) and below by \(r = 2 \, \cos \theta\) is
\[\begin{align*} V &= \iint_D f(r, \theta) \,r \, dr \, d\theta \\&= \int_{\theta=0}^{\theta=\pi} \int_{r=0}^{r=2 \, \cos \, \theta} (4 - r^2) \,r \, dr \, d\theta\\ &= \int_{\theta=0}^{\theta=\pi}\left.\left[4\frac{r^2}{2} - \frac{r^4}{4}\right|_0^{2 \, \cos \, \theta}\right]d\theta \\ &= \int_0^{\pi} [8 \, \cos^2\theta - 4 \, \cos^4\theta]\,d\theta \\&= \left[\frac{5}{2}\theta + \frac{5}{2} \sin \, \theta \, \cos \, \theta - \sin \, \theta \cos^3\theta \right]_0^{\pi} = \frac{5}{2}\pi\; \text{units}^3. \end{align*}\]
Notice in the next example that integration is not always easy with polar coordinates. Complexity of integration depends on the function and also on the region over which we need to perform the integration. If the region has a more natural expression in polar coordinates or if \(f\) has a simpler antiderivative in polar coordinates, then the change in polar coordinates is appropriate; otherwise, use rectangular coordinates.
Example \(\PageIndex{5A}\): Finding a Volume Using a Double Integral
Find the volume of the region that lies under the paraboloid \(z = x^2 + y^2\) and above the triangle enclosed by the lines \(y = x, \, x = 0\), and \(x + y = 2\) in the \(xy\)-plane.
Solution
First examine the region over which we need to set up the double integral and the accompanying paraboloid.
The region \(D\) is \(\{(x,y)\,|\,0 \leq x \leq 1, \, x \leq y \leq 2 - x\}\). Converting the lines \(y = x, \, x = 0\), and \(x + y = 2\) in the \(xy\)-plane to functions of \(r\) and \(\theta\) we have \(\theta = \pi/4, \, \theta = \pi/2\), and \(r = 2 / (\cos \, \theta + \sin \, \theta)\), respectively. Graphing the region on the \(xy\)- plane, we see that it looks like \(D = \{(r, \theta)\,|\,\pi/4 \leq \theta \leq \pi/2, \, 0 \leq r \leq 2/(\cos \, \theta + \sin \, \theta)\}\).
Now converting the equation of the surface gives \(z = x^2 + y^2 = r^2\). Therefore, the volume of the solid is given by the double integral
\[\begin{align*} V &= \iint_D f(r, \theta)\,r \, dr \, d\theta \\&= \int_{\theta=\pi/4}^{\theta=\pi/2} \int_{r=0}^{r=2/ (\cos \, \theta + \sin \, \theta)} r^2 r \, dr d\theta \\ &= \int_{\pi/4}^{\pi/2}\left[\frac{r^4}{4}\right]_0^{2/(\cos \, \theta + \sin \, \theta)} d\theta \\ &=\frac{1}{4}\int_{\pi/4}^{\pi/2} \left(\frac{2}{\cos \, \theta + \sin \, \theta}\right)^4 d\theta \\ &= \frac{16}{4} \int_{\pi/4}^{\pi/2} \left(\frac{1}{\cos \, \theta + \sin \, \theta} \right)^4 d\theta \\&= 4\int_{\pi/4}^{\pi/2} \left(\frac{1}{\cos \, \theta + \sin \, \theta}\right)^4 d\theta. \end{align*}\]
As you can see, this integral is very complicated. So, we can instead evaluate this double integral in rectangular coordinates as
\[V = \int_0^1 \int_x^{2-x} (x^2 + y^2) \,dy \, dx. \nonumber \]
Evaluating gives
\[\begin{align*} V &= \int_0^1 \int_x^{2-x} (x^2 + y^2) \,dy \, dx \\&= \int_0^1 \left.\left[x^2y + \frac{y^3}{3}\right]\right|_x^{2-x} dx\\ &= \int_0^1 \frac{8}{3} - 4x + 4x^2 - \frac{8x^3}{3} \,dx \\ &= \left.\left[\frac{8x}{3} - 2x^2 + \frac{4x^3}{3} - \frac{2x^4}{3}\right]\right|_0^1 \\&= \frac{4}{3} \; \text{units}^3. \end{align*}\]
To answer the question of how the formulas for the volumes of different standard solids such as a sphere, a cone, or a cylinder are found, we want to demonstrate an example and find the volume of an arbitrary cone.
Example \(\PageIndex{5B}\): Finding a Volume Using a Double Integral
Use polar coordinates to find the volume inside the cone \(z = 2 - \sqrt{x^2 + y^2}\) and above the \(xy\)-plane.
Solution
The region \(D\) for the integration is the base of the cone, which appears to be a circle on the \(xy\)-plane (Figure \(\PageIndex{10}\)).
We find the equation of the circle by setting \(z = 0\):
\[\begin{align*} 0 &= 2 - \sqrt{x^2 + y^2} \\ 2 &= \sqrt{x^2 + y^2} \\ x^2 + y^2 &= 4. \end{align*}\]
This means the radius of the circle is \(2\) so for the integration we have \(0 \leq \theta \leq 2\pi\) and \(0 \leq r \leq 2\). Substituting \(x = r \, \cos \theta\) and \(y = r \, \sin \, \theta\) in the equation \(z = 2 - \sqrt{x^2 + y^2}\) we have \(z = 2 - r\). Therefore, the volume of the cone is
\[\int_{\theta=0}^{\theta=2\pi} \int_{r=0}^{r=2} (2 - r)\,r \, dr \, d\theta = 2 \pi \frac{4}{3} = \frac{8\pi}{3}\; \text{cubic units.} \nonumber \]
AnalysisNote that if we were to find the volume of an arbitrary cone with radius \(\alpha\) units and height \(h\) units, then the equation of the cone would be \(z = h - \frac{h}{a}\sqrt{x^2 + y^2}\).
We can still use Figure \(\PageIndex{10}\) and set up the integral as
\[\int_{\theta=0}^{\theta=2\pi} \int_{r=0}^{r=a} \left(h - \frac{h}{a}r\right) r \, dr \, d\theta. \nonumber \]
Evaluating the integral, we get \(\frac{1}{3} \pi a^2 h\).
Exercise \(\PageIndex{5}\)
Use polar coordinates to find an iterated integral for finding the volume of the solid enclosed by the paraboloids \(z = x^2 + y^2\) and \(z = 16 - x^2 - y^2\).
- Hint
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Sketching the graphs can help.
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\[V = \int_0^{2\pi} \int_0^{2\sqrt{2}} (16 - 2r^2) \,r \, dr \, d\theta = 64 \pi \; \text{cubic units.} \nonumber \]
Example \(\PageIndex{6B}\): Finding Area Between Two Polar Curves
Find the area enclosed by the circle \(r = 3 \, \cos \, \theta\) and the cardioid \(r = 1 + \cos \, \theta\).
Solution
First and foremost, sketch the graphs of the region (Figure \(\PageIndex{12}\)).
We can from see the symmetry of the graph that we need to find the points of intersection. Setting the two equations equal to each other gives
\[3 \, \cos \, \theta = 1 + \cos \, \theta. \nonumber \]
One of the points of intersection is \(\theta = \pi/3\). The area above the polar axis consists of two parts, with one part defined by the cardioid from \(\theta = 0\) to \(\theta = \pi/3\) and the other part defined by the circle from \(\theta = \pi/3\) to \(\theta = \pi/2\). By symmetry, the total area is twice the area above the polar axis. Thus, we have
\[A = 2 \left[\int_{\theta=0}^{\theta=\pi/3} \int_{r=0}^{r=1+\cos \, \theta} 1 \,r \, dr \, d\theta + \int_{\theta=\pi/3}^{\theta=\pi/2} \int_{r=0}^{r=3 \, \cos \, \theta} 1\,r \, dr \, d\theta \right]. \nonumber \]
Evaluating each piece separately, we find that the area is
\[A = 2 \left(\frac{1}{4}\pi + \frac{9}{16} \sqrt{3} + \frac{3}{8} \pi - \frac{9}{16} \sqrt{3} \right) = 2 \left(\frac{5}{8}\pi\right) = \frac{5}{4}\pi \, \text{square units.} \nonumber \]
Exercise \(\PageIndex{6}\)
Find the area enclosed inside the cardioid \(r = 3 - 3 \, \sin \theta\) and outside the cardioid \(r = 1 + \sin \theta\).
- Hint
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Sketch the graph, and solve for the points of intersection.
- Answer
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\[A = 2 \int_{-\pi/2}^{\pi/6} \int_{1+\sin \, \theta}^{3-3\sin \, \theta} \,r \, dr \, d\theta = \left(8 \pi + 9 \sqrt{3}\right) \; \text{units}^2 \nonumber \]
Example \(\PageIndex{7}\): Evaluating an Improper Double Integral in Polar Coordinates
Evaluate the integral
\[\iint_{R^2} e^{-10(x^2+y^2)} \,dx \, dy. \nonumber \]
Solution
This is an improper integral because we are integrating over an unbounded region \(R^2\). In polar coordinates, the entire plane \(R^2\) can be seen as \(0 \leq \theta \leq 2\pi, \, 0 \leq r \leq \infty\).
Using the changes of variables from rectangular coordinates to polar coordinates, we have
\[\begin{align*} \iint_{R^2} e^{-10(x^2+y^2)}\,dx \, dy &= \int_{\theta=0}^{\theta=2\pi} \int_{r=0}^{r=\infty} e^{-10r^2}\,r \, dr \, d\theta = \int_{\theta=0}^{\theta=2\pi} \left(\lim_{a\rightarrow\infty} \int_{r=0}^{r=a} e^{-10r^2}r \, dr \right) d\theta \\ &=\left(\int_{\theta=0}^{\theta=2\pi}\right) d\theta \left(\lim_{a\rightarrow\infty} \int_{r=0}^{r=a} e^{-10r^2}r \, dr \right) \\ &=2\pi \left(\lim_{a\rightarrow\infty} \int_{r=0}^{r=a} e^{-10r^2}r \, dr \right) \\ &=2\pi \lim_{a\rightarrow\infty}\left(-\frac{1}{20}\right)\left(\left. e^{-10r^2}\right|_0^a\right) \\ &=2\pi \left(-\frac{1}{20}\right)\lim_{a\rightarrow\infty}\left(e^{-10a^2} - 1\right) \\ &= \frac{\pi}{10}. \end{align*}\]
Exercise \(\PageIndex{7}\)
Evaluate the integral
\[\iint_{R^2} e^{-4(x^2+y^2)}dx \, dy. \nonumber \]
- Hint
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Convert to the polar coordinate system.
- Answer
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\(\frac{\pi}{4}\)
Key Concepts
- To apply a double integral to a situation with circular symmetry, it is often convenient to use a double integral in polar coordinates. We can apply these double integrals over a polar rectangular region or a general polar region, using an iterated integral similar to those used with rectangular double integrals.
- The area \(dA\) in polar coordinates becomes \(r \, dr \, d\theta\).
- Use \(x = r \, \cos \, \theta, \, y = r \, \sin \, \theta\), and \(dA = r \, dr \, d\theta\) to convert an integral in rectangular coordinates to an integral in polar coordinates.
- Use \(r^2 = x^2 + y^2\) and \(\theta = tan^{-1} \left(\frac{y}{x}\right)\) to convert an integral in polar coordinates to an integral in rectangular coordinates, if needed.
- To find the volume in polar coordinates bounded above by a surface \(z = f(r, \theta)\) over a region on the \(xy\)-plane, use a double integral in polar coordinates.
Key Equations
- Double integral over a polar rectangular region \(R\)
\[\iint_R f(r, \theta) dA = \lim_{m,n\rightarrow\infty}\sum_{i=1}^m \sum_{j=1}^n f(r_{ij}^*, \theta_{ij}^*) \Delta A = \lim_{m,n\rightarrow\infty}\sum_{i=1}^m \sum_{j=1}^nf(r_{ij}^*,\theta_{ij}^*)r_{ij}^*\Delta r \Delta \theta \nonumber \]
- Double integral over a general polar region
\[\iint_D f(r, \theta)\,r \, dr \, d\theta = \int_{\theta=\alpha}^{\theta=\beta} \int_{r=h_1(\theta)}^{r_2(\theta)} f (r,\theta) \,r \, dr \, d\theta \nonumber \]
Glossary
- polar rectangle
- the region enclosed between the circles \(r = a\) and \(r = b\) and the angles \(\theta = \alpha\) and \(\theta = \beta\); it is described as \(R = \{(r, \theta)\,|\,a \leq r \leq b, \, \alpha \leq \theta \leq \beta\}\)
As with rectangular coordinates, we can also use polar coordinates to find areas of certain regions using a double integral. As before, we need to understand the region whose area we want to compute. Sketching a graph and identifying the region can be helpful to realize the limits of integration. Generally, the area formula in double integration will look like
\[\text{Area of} \, A = \int_{\alpha}^{\beta} \int_{h_1(\theta)}^{h_2(\theta)} 1 \,r \, dr \, d\theta. \nonumber \]
Example \(\PageIndex{6A}\): Finding an Area Using a Double Integral in Polar Coordinates
Evaluate the area bounded by the curve \(r = \cos \, 4\theta\).
Solution
Sketching the graph of the function \(r = \cos \, 4\theta\) reveals that it is a polar rose with eight petals (see the following figure).
Using symmetry , we can see that we need to find the area of one petal and then multiply it by 8. Notice that the values of \(\theta\) for which the graph passes through the origin are the zeros of the function \(\cos \, 4\theta\), and these are odd multiples of \(\pi/8\). Thus, one of the petals corresponds to the values of \(\theta\) in the interval \([-\pi/8, \pi/8]\). Therefore, the area bounded by the curve \(r = \cos \, 4\theta\) is
\[\begin{align*} A &= 8 \int_{\theta=-\pi/8}^{\theta=\pi/8} \int_{r=0}^{r=\cos \, 4\theta} 1\,r \, dr \, d\theta \\ &= 8 \int_{\theta=-\pi/8}^{\theta=\pi/8}\left.\left[\frac{1}{2}r^2\right|_0^{\cos \, 4\theta}\right] d\theta \\ &= 8 \int_{-\pi/8}^{\pi/8} \frac{1}{2} \cos^24\theta \, d\theta \\&= 8\left. \left[\frac{1}{4} \theta + \frac{1}{16} \sin \, 4\theta \, \cos \, 4\theta \right|_{-\pi/8}^{\pi/8}\right] \\&= 8 \left[\frac{\pi}{16}\right] = \frac{\pi}{2}\; \text{units}^2. \end{align*}\]


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5: Structure and Function of Plasma Membranes
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5.1: Components and Structure - Components of Plasma Membranes
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The plasma membrane protects the cell from its external environment, mediates cellular transport, and transmits cellular signals.
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5.2: Components and Structure - Fluid Mosaic Model
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The fluid mosaic model describes the plasma membrane structure as a mosaic of phospholipids, cholesterol, proteins, and carbohydrates.
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5.3: Components and Structure - Membrane Fluidity
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The mosaic nature of the membrane, its phospholipid chemistry, and the presence of cholesterol contribute to membrane fluidity.
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5.4: Passive Transport - The Role of Passive Transport
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Passive transport, such as diffusion and osmosis, moves materials of small molecular weight across membranes.
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5.5: Passive Transport - Selective Permeability
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The hydrophobic and hydrophilic regions of plasma membranes aid the diffusion of some molecules and hinder the diffusion of others.
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5.6: Passive Transport - Diffusion
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Diffusion is a process of passive transport in which molecules move from an area of higher concentration to one of lower concentration.
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5.7: Passive Transport - Facilitated Transport
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Facilitated diffusion is a process by which molecules are transported across the plasma membrane with the help of membrane proteins.
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5.8: Passive Transport - Osmosis
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Osmosis is the movement of water through a semipermeable membrane according to the concentration gradient of water across the membrane, which is inversely proportional to the concentration of solutes. Semipermeable membranes, also termed selectively permeable membranes or partially permeable membranes, allow certain molecules or ions to pass through by diffusion.
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5.9: Passive Transport - Tonicity
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Tonicity describes how an extracellular solution can change the volume of a cell by affecting osmosis. A solution’s tonicity often directly correlates with the osmolarity of the solution. Osmolarity describes the total solute concentration of the solution. A solution with low osmolarity has a greater number of water molecules relative to the number of solute particles; a solution with high osmolarity has fewer water molecules with respect to solute particles.
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5.10: Active Transport - Electrochemical Gradient
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To move substances against the membrane’s electrochemical gradient, the cell utilizes active transport, which requires energy from ATP.
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5.11: Active Transport - Primary Active Transport
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The sodium-potassium pump maintains the electrochemical gradient of living cells by moving sodium in and potassium out of the cell.
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5.12: Active Transport - Secondary Active Transport
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In secondary active transport, a molecule is moved down its electrochemical gradient as another is moved up its concentration gradient.
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5.13: Bulk Transport - Endocytosis
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Endocytosis is a type of active transport that moves particles, such as large molecules, parts of cells, and even whole cells, into a cell. There are different variations of endocytosis, but all share a common characteristic: the plasma membrane of the cell invaginates, forming a pocket around the target particle. The pocket pinches off, resulting in the particle being contained in a newly-created intracellular vesicle formed from the plasma membrane.
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5.14: Bulk Transport - Exocytosis
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Exocytosis is the process by which cells release particles from within the cell into the extracellular space.


4: The Second Law
The second law of thermodynamics concerns entropy and the spontaneity of processes. This chapter discusses theoretical aspects and practical applications.
We have seen that the first law allows us to set up a balance sheet for energy changes during a process, but says nothing about why some processes occur spontaneously and others are impossible. The laws of physics explain some spontaneous changes. For instance, unbalanced forces on a body cause acceleration, and a temperature gradient at a diathermal boundary causes heat transfer. But how can we predict whether a phase change, a transfer of solute from one solution phase to another, or a chemical reaction will occur spontaneously under the existing conditions? The second law provides the principle we need to answer these and other questions—a general criterion for spontaneity in a closed system.


16.6: Low Wages and the Deserving Worker
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So far in this chapter we have focused on food consumption, on what people eat. But how that food is produced and exchanged is important in its own right. Indeed, if we include all the jobs involved in providing food – from farming and fishing through processing and distribution, right up to retailing and cooking – then it is arguably the most important income-generating sector in the world. In the United States (US), there has been a long history of struggles over food work. John Steinbeck captured a slice of it in his 1939 book Grapes of Wrath , writing about a family of tenant farmers evicted from their home in Oklahoma and who end up working on a peach plantation in California for a pittance. This fictional book based on real events echoes in the lives of farmworkers in the United States today. Jobs like picking fruit and weeding vegetables are still tough and still done by migrants – only now they typically come from Latin America. In 2012, their average pay was less than $19,000 a year. The US government’s own statistics would place this income thousands of dollars below the minimum threshold for meeting the basic needs of a family of four. In other words, even though they were living in the world’s richest nation, they were living in relative poverty.
There are some differences between Steinbeck’s story and contemporary events, though. In Grapes of Wrath , a preacher called Casy tries to organise his fellow workers into a trade union and is murdered by the police for his troubles. For the Coalition of Immokalee Workers, a group of immigrant tomato pickers based in Immokalee Florida, their initial meetings in a local church grew into something much bigger. They first used tactics like work stoppages and hunger strikes to demand higher wages from their employers, but as their public profile grew they sought to reorganise the food supply chain itself. In 2011 the Coalition launched the Fair Food Program. Major restaurant and supermarket chains were encouraged to pay a few cents more for a pound of tomatoes and to buy these tomatoes from suppliers who pledged to follow labour law and put the extra money in their workers’ wage packets. The Coalition scored its biggest success when the biggest retailer in the world, Walmart, agreed to join the Fair Food Program and to extend it beyond just tomatoes.
But while Walmart made commitments to these workers, with its own workers it has been less forthcoming. In 2012 its regular employees like cashiers, cleaners and warehouse assistants were paid on average just $8.81 an hour (Buchheit 2013). This meant that they, too, were paid a poverty wage and thus qualified for additional social security benefits like food vouchers, many of which were then spent by workers back in Walmart stores! This costs the government billions per year and is surely the grand paradox of the American economy. For all its wealth and Wall Street millionaires, the national minimum wage is so low that many people in full-time work still cannot make ends meet. Nor is it just Walmart where this happens. Supermarket cashiers, farm labourers, fast food servers, cooks, dishwashers, bartenders and waiting staff are all among America’s lowest paid workers. The price of cheap food in the country has been gross inequality.
In both the Walmart case and that of the Coalition of Immokalee Workers, the position of the deserving worker has been crucial in contesting this inequality. We can see this first in the way that immigration policy has been conducted. For years US farm companies lobbied the government to allow them access to cheap foreign labour, which the government achieved by issuing temporary immigration visas and turning a blind eye to the use of additional undocumented workers. This created tensions with the general public, some of whom were worried about wages being undercut and others about the decline of ‘American values’. A 2013 proposal by Republican and Democratic Party senators to offer permanent citizenship to undocumented farm workers thus had to cast them in a particular light. They were not called ‘illegal immigrants’, as was more usual in political discourse, but portrayed as ‘individuals who … have been performing very important and difficult work to maintain America’s food supply’ (Plumer 2013). What the politicians were implying was that these were honest and hardworking people that could and should be made into Americans.
A second example is the way that trade unions have tried to organise Walmart employees across national borders. The company’s takeover of food retailers in other countries has given it a truly global workforce. Walmart now employs over two million people worldwide; only the United States and Chinese militaries employ more. Concerned that the labour standards in its American operations might be adopted in these supermarkets and their supply chains too, groups like the UNI Global Union have thus tried to link people together through the shared subjectivity of the deserving worker and create a sense of international solidarity between them. As a UNI coordinator put it: ‘When I can connect a Chinese worker with a Mexican worker then it doesn’t become about a Chinese worker taking their job. Workers can see, “Oh they [Walmart] are screwing us both. We have to unite to win”’ (Jackson 2014).


9.3: How can we teach to meet all students' needs?
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By Honorina Maristela
Learning Objectives
- Define differentiated instruction and understand the use of this approach in the classroom.
- Identify the teacher’s role in differentiated instruction.
Introduction
In a traditional classroom setting, the teachers talk while the students listen. Nowadays, traditional settings have completely changed in the field of education. The way the classroom are set-up, the teaching and learning style, the accommodation of individual students and application of small group instruction made the school and teaching challenging both for teachers and students.In addition to these changes, differentiated instruction has been emphasized.
The district and state and federal governments have established our standards and have handed our curriculum down to us. These standards make up goals established for all students. How we reach the goals may require different paths. The core of differentiated instruction is flexibility in content, process, and product based on students' strengths, needs and learning styles. (Levy, 2008 cited in The Clearing House, 2008)
What is Differentiated Instruction?
At the most basic level, differentiating instruction means “shaking up” what goes on in the classroom. Students have multiple options for taking in information, making sense of ideas and expressing what they learn. (Tomlinson, 2001) In other words, differentiated instructing is catering to student’s needs and interests so they can acquire learning efficiently..
The State of Virginia encourages the school districts to apply differentiated instruction in the classrooms. In 2004, the Superintendent of Public Instruction conducted a Professional Development to all School Heads, member of Virginia Association of Elementary School Principals, Assistant Principals and Lead teachers to participate in the conference entitled “ Differentiated Instruction: Meeting the Needs of All Children” the highlight for this conference included the special needs students and at-risk students.
Why differentiate?
Individual students are different in many ways. The use of differentiated instruction will gives the students the opportunity to meet their needs in an intensive step that will expedite the learning skill of the individual student. We as teachers give the support and the careful planning of the instruction to make sure that the activities will be in the students’ instructional level and interest.
Differentitate Why: addresses the teachers’ reason for modifying the leaning experience. Teachers believe modification is important for many reasons. Three key reasons include access to learning, motivation to learn and efficiency of learning. Any or all of these three reasons for differentiating instruction can be tied to students readiness, interest, and learning profile. (Tomlinson, 1999)
In differentiated classroom, teachers begin at the level of the students, not the front of a curriculum guide. They accept and build upon the premise that learners differ in important ways. Thus, they also accept and act on the premise that teachers must be ready to engage students in instruction though different learning modalities, by appealing to differing interest and by using varied rates of instruction along with varied degrees of complexity. In differentiated classrooms, teachers ensure that a student competes agains himself or herself as he or she grows and develops more than he or she competes against other students. (Tomlinson, 1999)
Though differentiated instruction and activities, students take a greater responsibility and ownership of their own learning, activities that are primarily focused on students’ multiple intelligences, higher order thinking, and learning styles. With this in mind, differentiated instruction is an effective tool to implement in the classroom in order to meet students’ learning style and strengthen multiple intelligences and encourage them to use high order thinking. (Bailey and Williams-Black, p. 134)
How to Differentiate?
It has been a big question for teachers how to differentiate instruction in the classroom. It is indeed a challenging task to put into use this approach. I myself can relate to this for it took me five years to attain the effectiveness of this approach and I am still learning until now. The preparation and designed activities should target the individual needs of the students.
Being in an inclusion is a tough job, having four groups of students with individual needs ranging from high, average, low level and special needs students with IEP (Individualized Education Program) and 504 plans. It is a team work of the teacher, special education teacher and the students in carrying out the activities in the classroom.
Use formal and informal assessments in making decision with small group instruction. Most schools are required certain formal assessments such as TPRI(Texas Primary Reading Assessment, DIBLE(Dynamic Indicators of Basic Early Literacy Skill, DRA( Developmental Reading Assessment) and/or QRI(Qualitative Reading Inventory. (Diller, 2007)
These assessments are useful tools for teachers for figuring out how to start with their small groups. Once the teacher determined the level of the individual students she may begin applying the differentiated instruction in the classroom. The teacher can design her own activities that are according to the instructional level of the students. One great example is in reading. Using leveled books that are color coded are useful in targeting the instructional level of the students and work towards it to improve to the fullest. The idea of using the flexible grouping of Debbie Diller is useful for teachers to differentiate. As Diller stated, to create flexible group folders for thinking about small groups as well as in whole group. Once the student show progress he/she can be moved into different group. (Diller, 2007,p. 24)
Flexible groupings were defined by Radencich and McKay (1995) as” grouping that is not static, where members of the group change frequently. This emerged as a practice to address concerns. It acknowledges that all grouping patterns- large groups, small groups, teams and individuals have value because they all offer the reader slightly different experiences with different outcomes. Students may work with a partner, in a small cooperative or teacher-led group, or with the whole class. The basis for this may be students’ interest or needs. ( Ford, 2005 cited in Radencich and McKay, 1995)
In planning flexible grouping teachers should consider the strengths and weaknesses of each grouping approach and then put them together to allow the teacher to best meet the needs of the classroom. The groups are formed and dissolved as needs changes to allow for maximum flexibility, avoiding the static nature of the grouping patterns of the past. ( Ford, 2005 cited in Radencich and McKay, 1995)
Note
Key Principles of a Differentiated Instruction
- The teacher is clear about what matters in subject matter.
- The teacher understands, appreciates, and builds upon student differences.
- Assessment and instruction are inseparable.
- The teacher adjusts content, process and product in response to student readiness, interest, and learning profile.
- All students participate in respectful work.
- Students and teachers are collaborators in learning.
- Goals of a differentiated classroom are maximum growth and individual success.
- Flexibility is the hallmark of a differentiated classroom.( Tomlinson, 1999)
Teachers Role
Mixed-ability classrooms that offer differentiated instruction make good sense for teachers, as well a student. For many teachers, offering differentiated instruction first requires a paradigm shift. ( Tomlinson, 2001)
Teachers who become comfortable with differentiated classroom would probably say their role differs in some significant ways from that of a traditional teacher. When teacher differentiate instruction, they move away from being themselves as keepers and dispensers of knowledge and move toward seeing themselves as organizers of learning opportunities. (Tomlnison, 2001)
Teachers who differentiate instruction focus on their role as coach or mentor(Tomlinson, 2001). We serve as a guide and facilitator for the students in doing their responsibilities in completing their task. We organize the activities for them to discover, explore and share acquired knowledge. ..
Note
A teacher is a compass that activates the magnets of curiosity, knowledge, and wisdom in the pupils. ~Ever Garrison
Conclusion
When applying differentiated instruction time is required in order for it to work. It is not an overnight activity to see the effectiveness of this approach. It takes time and dedication to help the learners be successful.
For most teachers, the classroom is the place where we spend our career. The classroom is the place where we give the better part of our lifetime trying to make a difference. It is a curiosity of teaching that no two days are the alike. We must remember that we have every opportunity to transform ourselves and our practice, just as we have every opportunity to stagnate, remain much the same teacher we were when we began. ( Tomlinson, 1999)
References
Bailey, J.P. and Williams-Black,T.H.(2008).Differentiated instruction three teacher's perspective.College Reading Association Yearbook(2008).pp. 29,134. Retrieved January 30, 2009 from http://search.ebscohost.com.proxy.lib.odu.edu/login.aspx?direct=true&db=ehh&AN=33435085&site=ehost-live
Diller, D. (2007).Making the most of small groups differentiation for all.Markham,Ontario Canada:Pembroke Publishes,pp. 21–24.
Ford, M.(2005).Differentiation through flexible grouping successfully reading all readers.Learning Point Associates(2005),p. 3.Retrieved February 7, 2009 from www.cpsgifted.org/pdf/flexibleGrouping1.pdf
Levy, H.(2008).Meeting the needs of all students through differentiated instruction:Helping every child reach and exceed standtards.Clearing House;Mar/Apr2008.Vol.81(4),2.Retrieved from February 6, 2009 http://web.ebscohost.com.proxy.lib.odu.edu/ehost/dilivery?vid=7&hid=113&sid=1cb60ea3 ...
Tomlinson,C.A. (1999).The differentiated classroom responding to the needs of all learners.Alexandria VA:Association for Supervision and Curriculum Development Publications,pp. 2,48-49,119.
Tomlinson, C.A. (2001).How to differentiate instruction in mixed ability classrooms(2nd ed.)Alexandria VA:Association for Supervision and Curriculum Development Publications,p. 16.
Common Wealth of Virginia.Department of Education.Retrieved February 5, 2009 from www.doe.virginia.gov/VDOE/suptsmemos/2004/inf025.html
Special Needs- Differentiation in Action.Retrieved February 8, 2009 from http://www.teachers.tv/video/21992


7.3: u-substitution
Recall the chain rule:
\(\boxed{\cfrac{d}{dx} f(g) = f'(g) \cdot g'}\)
The method of “\(u\)-substitution” is a way of doing integral problems that undo the chain rule. It also helps deal with constants that crop up.
- Identify an “inside” function whose derivative is multiplied on the outside, possibly with a different constant. Call this “inside” function \(u\).
- Compute \(\frac{du}{dx}\) and solve for \(dx\).
- Use substitution to replace \(x \to u\) and \(dx \to du\), and cancel any remaining \(x\) terms if possible.
- Integrate with respect to \(u\). If at this point you still have any \(x\)s in your problem, either you made a mistake or the method of \(u\)-substitution will not work for this problem.
- Substitute back the \(x\)s back into the answer before evaluating the definite integral.
Let’s do some examples.
We will follow the steps of \(u\) substitution.
- In this case, the “inside function” is \(u = -5x\).
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If we compute \(\frac{du}{dx}\), we see the derivative of \(-5x\) is \(-5\). Hence \(\frac{du}{dx} = -5,\) and we have solving for \(dx\)
\[\begin{align*} \frac{du}{dx} & = -5 \\ du & = -5 dx \\ -\frac{1}{5} du & = dx \end{align*}\]
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Going back to the original problem and using substitution \(u = -5x\) and \(dx = -\frac{1}{5} du\), we thus have:
\[\begin{align*} \int_0^3 e^{-5x}dx & \int_0^3 e^u dx \\ & = \int_0^3 e^u \left(- \frac{1}{5} du\right) \\ & = \int_0^3 -\frac{1}{5} e^udu. \end{align*}\]
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Integrating with respect to \(u\),
\[\begin{align*} \int_0^3 - \frac{1}{5} e^udu & = -\frac{1}{5} \int_0^3 e^udu \\ & = -\frac{1}{5} e^u \Big|_0^3 \end{align*}\]
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We now substitute \(u = -5x\). We have
\[\begin{align*} -\frac{1}{5} e^u \Big|_0^3 & = -\frac{1}{5}e^{-5x} \Big|_0^3 \\ & = \left(-\frac{1}{5}e^{-15}\right) - \left(-\frac{1}{5}e^{-5(0)} \right) \\ & = -\frac{e^{-15}}{5} + \frac{1}{5} \\ & = \frac{1 - e^{-15}}{5}\\ & \approx \boxed{0.2} \end{align*}\]
There we go.
Again, we will go through the steps of \(u\)-substitution.
- The inside function in this case is \(x^2 + 1\). We can see that the derivative is \(2x\), and this is good since there is an \(x\) multiplied out in front (the \(2\) is just a constant we can deal with.) Set \(u = x^2 + 1\).
- We see \(\frac{du}{dx} = 2x\), and hence solving for \(dx\) we have \(\frac{du}{2x} = dx\).
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Subbing in \(u = x^2 + 1\) and \(dx = \frac{du}{2x}\), we have
\[\begin{align*} \int x (x^2 + 1)^7dx & = \int x u^7dx \\ & = \int x u^7 \left( \frac{du}{2x} \right) \\ & = \int \frac{x}{2x} u^7du \\ & = \int \frac{1}{2} u^7du \end{align*}\]
Great, the \(x\)s are all gone!
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We can now integrate with respect to \(u\).
\[\begin{align*} \int \frac{1}{2} u^7du & = \frac{1}{2} \int u^7du \\ & = \frac{1}{2} \frac{u^8}{8} \\ & = \frac{u^8}{16} \end{align*}\]
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Finally, we sub in the \(x\)s again using \(u = x^2 + 1\).
\[\begin{align*} \frac{u^8}{16} & = \boxed{\frac{(x^2 + 1)^8}{16} + C} \end{align*}\]
Since this is an indefinite integral, we add the constant of integration.
Again, we will go through the steps of \(u\)-substitution.
- The inside function in this case is \(\ln(x)\). We can see that the derivative is \(\frac{1}{x}\), and this is good since there is an \(x\) dividing the rest of the problem. Set \(u = \ln(x)\).
- We see \(\frac{du}{dx} =\frac{1}{x}\), and hence solving for \(dx\) we have \(\frac{du}{\frac{1}{x}} = dx\).
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Subbing in \(u =\ln(x)\) and \(dx = \frac{du}{\frac{1}{x}}\), we have
\[\begin{align*} \int \frac{8 (\ln(x))^3}{x}dx & = \int \frac{8 u^3}{x} \cdot \frac{du}{\frac{1}{x}} \\ & = \int \frac{8 u^3}{x \cdot \frac{1}{x}}du \\ & = \int \frac{8 u^3}{1}du \\ & = \int 8 u^3du \\ \end{align*}\]
Great, the \(x\)s are all gone!
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We can now integrate with respect to \(u\).
\[\begin{align*} \int 8 u^3du & = 8 \cdot \frac{1}{4} u^4 \\ & = 2 u^4 \end{align*}\]
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Finally, we sub in the \(x\)s again using \(u = \ln(x)\).
\[\begin{align*} \int \frac{8(\ln(x))^3}{x}dx & = \boxed{2 (\ln(x))^4 + C}. \end{align*}\]
If the inside function is linear, the \(u\)-substitution is much simpler, and there is even a formula for it (just like in the \(\int e^{-5x}dx\) example above). By the chain rule with \(g = mx + b\) and \(g' = m\), we have
\[\begin{align*} \frac{d}{dx} \frac{1}{m} f(mx + b) & = \frac{1}{m} \frac{d}{dx} f(mx + b) \\ & = \frac{1}{m} \left( f'(g) \cdot g' \right) \\ & = \frac{1}{m} f'(mx + b) \cdot m \\ & = \frac{m}{m} f'(mx + b) \\ & = f'(mx + b) \end{align*}\]
If we integrate both sides of this equation, we have the following useful rule which I call the “chain rule shortcut”:
\(\boxed{\int f'(mx + b)dx = \frac{1}{m} f(mx + b)}\)
This is especially important if \(f(x) = e^{mx}\). In this case, we have
\(\boxed{\int e^{mx}dx = \frac{1}{m} e^{mx}}\)
In other words, you integrate just like normal without any \(u\)-substitution, and then add a \(\frac{1}{m}\) factor for the fact that you have \(mx + b\) inside the function instead of just an \(x\).
Let’s see a couple of examples of this:
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Find \(\int (6x - 3)^4dx\).
Using the inverse power rule, we see this becomes \(\frac{(6x - 3)^5}{5}\). However, we need that \(\frac{1}{m}\) factor since the problem has a \(6x - 3\) in it. So the final answer is
\(\frac{1}{6} \frac{(6x - 3)^5}{5} = \frac{(6x - 3)^5}{30} + C.\)
You’d get the same thing doing a full \(u\)-substitution with \(u = 6x - 3\). This way, though, you save some time by just multiplying by \(\frac{1}{6}\).
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Find \(\int_{-2}^{-1} (3x + 5)^7dx\).
Again, we use the \(u\)-sub shortcut — we just need to do power rule and remember a \(\frac{1}{m}\) factor, which in this case is \(\frac{1}{3}\).
\[\begin{align*} \int_{-2}^{-1} (3x + 5)^7dx & = \frac{1}{3} \frac{(3x + 5)^8}{8} \Big|_{-2}^{-1} \\ & = \frac{(3x + 5)^8}{24} \Big|_{-2}^{-1} \\ & = \left( \frac{(3(-1) + 5)^8}{24} \right) - \left( \frac{(3(-2) + 5)^8}{24} \right) \\ & = \frac{256}{24} - \frac{1}{24} \\ & = \frac{255}{24} \\ & = \frac{83}{8} \end{align*}\]
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Find \(\int e^{0.05t}dt\).
The antiderivative here is just \(\frac{1}{0.05} e^{0.05t}\). We can plug \(20\), so the answer is \(\boxed{20 e^{0.05t} + C}\).