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Chapter Sixteen Objectives
By the end of this chapter, you will be able to:
- Explain how large, complicated problems can be broken down into smaller solvable ones
- Identify the importance of studying and bringing about change in observable behaviors
- Understand how behavioral approaches have been used in Community Psychology
You may have heard of B. F. Skinner , who pioneered the use of behavioral approaches to show how rewards and punishments can change the behaviors of animals and people. This behavioral revolution has had a tremendous effect on psychology in terms of how therapy is now practiced in countries throughout the world. The old Freudian perspective has been replaced by a variety of behavioral-type therapies, used to help patients with a wide variety of common problems such as anxiety and depressive disorders. Most behavioral studies have been conducted in hospitals in inpatient settings or other types of institutions with much less behavioral research occurring in community settings such as homes or neighborhoods (Hanley et al., 2003). When behavioral-type work is conducted in community settings and has the values from the field of Community Psychology (as reviewed in Chapter 1, Jason, Glantsman, O’Brien, & Ramian, 2019), it has been referred to as Behavioral Community Psychology .
As we will see in this chapter, the Behavioral Community Psychology approach allows for a more focused and targeted method to define and change a range of community and social problems. In a sense, this approach breaks down large and complex problems into smaller ones that are much more manageable to solve. By specifying and measuring specific behaviors, such as littering or tobacco use, behavioral community psychologists can chart those behaviors over time to determine whether or not a particular intervention is effective. We will show in this chapter how these types of practical and easy-to-implement interventions can lead to what are called small wins , which can be reinforcing and sustain the work of community psychologists over time. We can change the world through being extremely strategic and focused by breaking down large problems into smaller ones that are solvable.
Ok, let’s give an example of a problem that seemed very difficult to solve. If you can believe it involved uncollected dog droppings in Chicago. There as a time when nobody knew how to even approach this problem, let alone study or solve it. But as you will see, by collecting data and showing that there was a possible solution, change did occur for not only Chicago but also other urban areas in the US. The behavioral community psychology Case Study 16.1 was carried out at a time when there were no formal laws that required dog owners to pick up after their dogs.
Case Study 16.1
Getting the Scoop on Behavioral Interventions
Leonard Jason invited his Alderman in Chicago to his community psychology class, and Jason said to himself that he would tackle whatever the Alderman mentioned as the most pressing community issue. When informed that it was uncollected dog feces, Jason’s mouth fell open in disbelief, but he pledged to work on this issue. Jason’s team first counted all fresh dog feces within an eight by five block area surrounding DePaul University. The fact that 1,147 droppings were within this area suggested that dog litter was a serious and prevalent problem within this community. Next, the team for five hours daily recorded the number of dogs, the number who defecated, and the number of dog defecations picked up by their owners. In addition, all defecations were picked up and weighed each morning. During the baseline phase, few dog owners were observed to pick up after their dogs and over 19 pounds of dog defecations were deposited in the target block. When dog owners were given instructions and a demonstration concerning how to use a plastic bag to pick-up dog feces, 82% of the dog owners proceeded to pick up after their dogs.
These findings indicated that the prompting intervention, which applied instructions and modeling, effectively motivated dog owners to dispose of their dogs’ waste properly.
At the study’s end, the Chicago alderman asked Jason to present the above data at City Hall in order to support a proposed ordinance which would require dog owners to have in their possession a pooper scooper when walking dogs. This ordinance was passed by the City Council, making Chicago one of the first cities in the country to pass a pooper scooper ordinance. The alderman to whom Jason had originally provided the data mentioned to him that this study, which received considerable media exposure, had helped change the politicians’ perception of the problem. The Alderman wrote to Jason: “In the past, this problem has often been scoffed at and not taken seriously. Your comments regarding the dog defecation problem altered that perception greatly.” Chicago’s ordinance became a model for other similar ordinances in towns around the country (Jason, 2013).
Although the subject matter of this study is a bit hard to believe, it is a true story. It shows the usefulness of listening to the community for ideas about what is important, even if it sometimes involves getting one’s hands dirty in the process. In other words, the subject matter of the intervention was decided upon by the needs of the community, and this is a critical part of behavioral community interventions. Also, by focusing on one small area and collecting data on dogs and their owners, it was possible to show that change could occur. In fact, it was found through the behavioral intervention that dog owners were willing to pick up after their dogs. Most importantly, the findings were transmitted to the legislative process, and a new ordinance came into law that made a real difference to the community. By getting into the trenches, getting our hands dirty, and being willing to study dog litter, we were able to influence dog owners to pick up. Just think what else is possible to accomplish.
HISTORICAL ROOTS
Some of the first developments of learning theory started in the 1890s when Ivan Pavlov demonstrated the importance of the environment on behavior. Pavlov wanted to measure the amount of salivation in dogs when they were given food, predicting that the dogs would begin to salivate in response to being fed. A researcher would bring food to the dogs whenever he walked into the lab. Pavlov noticed that after a while, the dogs began to salivate in response to hearing the researcher’s footsteps, meaning they began to associate the footsteps with being fed, which caused a behavioral response (e.g., salivation).
Pavlov (1897) examined this discovery further with a series of experiments , in which the dogs underwent numerous trials and were exposed to a sound that was immediately followed by food. Pavlov discovered that when he just made the sound that accompanied food, the dogs began to salivate even if the food was not presented. Through the systematic control of the environment (i.e., onset of a sound and delivery of food), the animals learned to associate the sound with the food that followed. This fundamental learning process is referred to as Classical Conditioning .
Later, B. F. Skinner further examined behavioral responses and developed Operant Conditioning through the use of small boxes that contained a lever that lab rats pressed to release food. In these experiments , a rat was placed in the box and then observed. At first, the rat scurried around and then accidentally pulled the lever that released the pellets of food. In successive trials, Skinner observed that the time in which it took the rat to release the pellets of food decreased each time it was placed in the box. The rat began to learn that pulling the lever resulted in the delivery of food. Skinner taught the rats to pull the lever by rewarding the behavior, which became the basis for Operant Conditioning . This theory builds upon the work of Pavlov in understanding the relationship between one’s environment and behavior. Skinner coined the term Behavioral Analysis (1953) to systematically demonstrate how the behavior of individuals and groups is influenced by the environment. His work led the way to understand how animal and human behavior changed through reinforcement and punishment.
These theories demonstrate the importance of the environment on shaping behavior. Environments have a powerful influence on individuals, and this work has provided insights for community psychologists as they try to understand larger and more complex ecological systems.
APPLIED BEHAVIOR ANALYSIS
Behavioral community psychologists often use the term Applied Behavior Analysis to understand and explain human behavior as a function of the interaction between behavior and the environment. Applied Behavior Analysis is rooted in Learning Theory , which focuses on objectively observable behaviors and how learning occurs when individuals respond to environmental factors or stimuli. Positive reinforcement and punishment can be used to teach and sustain behavior and shape individual as well as community behavior.
Another term that is often used in the behavioral literature is Behavior Modification , which refers to the systematic application of learning principles and techniques to assess and change an individual’s behavior (Martin & Pear, 2015). Often the terms Behavior Modification and Applied Behavior Analysis are used interchangeably within the field of Behavioral Community Psychology. Below are some terms used frequently in this field:
Positive Reinforcement leads to increases in the chances that a behavior will continue to occur . On the other hand, Punishment often leads to the reduction of the rate or likelihood of a behavior, which can result in the behavior being Extinguished.
Here is an example: Imagine that you see a person driving on the highway at a speed exceeding the local limit and the police give the person a speeding ticket. It is possible that this ticket (i.e., consequence) reduces the likelihood of future speeding behavior. So, the person’s behavior is changed by the consequences as they now drive at the speed limit.
Shaping of behavior involves establishing a complex set of behaviors using a set of steps. Teaching a child to cross a street involves teaching several behaviors connected in a certain order. This might look something like this: walk up to the corner, push the button to activate the crosswalk and traffic light, wait until you get the correct “walk” signal and the cars are stopped, and then quickly get across the street before the light changes.
Stimulus control highlights how characteristics of the environment might influence the occurrence of a behavior. To give an example of this principle, in Case Study 16.2 below, a group of Behavioral Community Psychology investigators examined the effects of altering traffic light pattern sequences on driver compliance at a busy, urban intersection in Chicago. In this case study, you will see that by making a small change to the traffic light pattern, car accidents were reduced.
Case Study 16.2
A Change in Lights: A Reduction in Car Accidents
This study began when a hazardous situation was observed with two sets of traffic lights that were separated by only about 100 feet. Drivers that were stopped at the first traffic light that was red, next saw this light turn green, and they began driving to the second traffic light which was in the process of turning from yellow to red. The drivers already had waited a long time for the first light to turn green, and were probably annoyed at facing another light that was going to delay them. As a consequence, more than half of the drivers sped right by the next light changing from yellow to red and this resulted in many accidents. Working with officials from the city of Chicago, the investigators then slightly altered light pattern sequences so that when the second set of traffic lights were approached, the lights were not changing from yellow to red. This small change resulted in almost all of the drivers not passing through a yellow or red light at the second traffic light. Following the completion of the study, the traffic engineer approved the permanent alteration of the light timing pattern that increased traffic rule compliance. Accident data collected before and after the light timing changes indicated a reduction in automobile accidents (Jason, Neal, & Marinakis, 1985).
ADDITIONAL BEHAVIORAL CONTRIBUTIONS
Several additional important developments in the behavioral approach have occurred. For example, Albert Bandura (1986) discovered that behavior is also learned from the environment through the process of observation (see the video below for a brief history of Bandura’s Bobo Doll Experiment).
A YouTube element has been excluded from this version of the text. You can view it online here: https://press.rebus.community/introductiontocommunitypsychology/?p=766
Bandura showed that it is possible to learn by just observing another person engaging in a particular behavior. So, a child might watch a parent use a fork to eat food, and then the child might engage in that behavior from just watching the parent engaging in the behavior. This work has helped scientists better understand how aggressive behavior can be learned by youth, and this can occur when youth observe violence being reinforced. In addition, Bandura helped us better understand the importance of the thoughts and feelings that intervene between the stimuli and responses. In a sense, we are active information processors , and therefore cognitive factors are also involved in learning.
A final development that has attracted considerable media attention is the emergence of the field of behavioral economics, which is derived from the fields of psychology and economics. In essence, we are susceptible to temptations and often make bad decisions even though there are better options. For example, it is not uncommon to eat high fat and high-calorie snacks rather than healthier foods, which has resulted in an obesity crisis in the US. Reed, Niileksela, and Kaplan (2013) suggest that this apparent irrationality can still be understood based on the principles of operant learning. Some risk-taking behaviors, such as unhealthy food choices, have more reinforcing value than less risk-taking behaviors such as healthy food choices.  Therefore, it is possible to understand responses such as these types of food choices (even though these foods can lead to obesity and other health complications) by the fact that these behaviors are being reinforced.
CONTRIBUTIONS OF BEHAVIORAL COMMUNITY PSYCHOLOGY
Most applications of Applied Behavior Analysis and Behavior Modification have been within the medical model , such as working one on one with patients having severe developmental disabilities. However, Behavioral Community Psychology interventions have addressed issues such as littering (Clark, Burgess, & Hendee, 1972); recycling (Zulas, 2009); illegal sales of cigarettes to minors (Jason, Ji, Anes, & Birkhead, 1991); drug addiction (Silverman, Roll, & Higgins, 2008); bullying in schools (Embry, 2002); blood donations (Ferrari & Jason, 1990); and identifying and conducting behavioral analyses on concerns of community members (Arellano et al., 2016; Balcazar, Garcia-Iriarte & Suarez-Balcazar, 2009; Suarez-Balcazar & Balcazar, 2016). Many of these interventions have embraced Community Psychology values summarized by Fawcett (1991), and include:
1) Valuing collaborative relationships
2) Focusing on behavior-environment relationships of importance to communities
3) Conducting research focusing on modifiable and sustainable environmental events
4) Involving setting and research measures relevant to community concerns
5) Using measurement systems that capture the dynamic relationship between behavior and environment
6) Developing interventions owned by the community and sustainable with local resources
7) Providing interventions focused on maximizing impact and benefits to the community
8) Spreading effective interventions
9) Communicating effectively to the community
stakeholders
10) Contributing to fundamental change. These values have guided the work of behavioral community psychologists (Glenwick & Jason, 1980).
Below are case studies that provide several examples of Behavioral Community Psychology interventions. Case Study 16.3 shows how behavioral community psychologists collaborated with community-based organizations to influence legislative officials to support laws that contributed to second-order change , in this case protecting the safety of infants and children. Before child car restraints became law, thousands of children in the US were either injured or killed in car accidents. In fact, the leading cause of death for children under one year of age was car accidents due to not being protected in infant seats.
Case Study 16.3
A Behavioral Community Psychology Approach: Changing Laws and Saving Lives
A national coalition of behavioral community psychologists, that included Leonard Jason and Stephen Fawcett, was trying to influence legislators to pass legislation that would require infants and children to wear a seatbelt or to be placed in an appropriate car or infant seat while in cars. In Illinois, Jason and his colleagues worked with a community-based organization that was trying to pass needed child restraint legislation. Their hunch was that data gathered on behavioral and self-report information could be used to influence the debate about passing needed legislation. For months, they looked inside cars to see whether or not infants and children were placed in car restraints. They also used telephone surveys to collect information about the public’s attitude toward the child-restraint bill. The goal was to use both data collection on attitudes and firsthand observations to build a more convincing case when trying to influence policy officials. Although they were also working closely with an Illinois organization that was advocating for the passage of this bill, they were unsure if the data would persuade legislators. They sent the collected information to a randomly selected half of the Illinois state legislators prior to a vote on the child passenger restraint bill. In that letter, the senators were informed that 140 children in Illinois were killed and 25,828 injured in automobile accidents over the last six years in Illinois. The authors also pointed out that, through their observations, 93% of Illinois children were not placed in adequate restraints while riding in cars. They also provided the legislators with the results of the survey, in which 78% of adults supported the child passenger restraint bill. By sending this critical information to half of the legislators, they were able to see whether the targeted letter had made a difference in encouraging support for this important legislation. Significantly more senators (79%) who received the information voted for passage of the bill, whereas only 53% of senators who did not receive the letter voted for the bill. Their intervention was a success, and even the governor requested a copy of the findings before finally signing the legislation. With the passage of the legislation, for children between the ages of one to four, the use of appropriate car restraints increased from 13% to 42%. Rates of appropriate restraints increased from 49% to 74% for infants less than one year of age. Most importantly, children’s deaths caused by traffic accidents decreased by 53%. Comments made by members of the Illinois Child Passenger Safety Association included: “The data were very, very interesting. It was a building block in the passage of the bill.” “Those who had the data and understood them, it made them more forceful and vocal in support of the bill” (Jason & Rose, 1984).
For many social and community problems, working at the policy level can have more substantial and enduring results, as the car restraint case study illustrated. Psychologists often provide therapy to those who are injured in car accidents, but there are clear advantages to using our more preventive behavioral community psychology strategies to change laws that influence parents to appropriately protect their children while being driven in cars.
The next case study illustrates how the community researchers used principles of Applied Behavior Analysis to study a concrete behavior—compliance with traffic signs by drivers–and developed operational definitions of terms to observe the behavior in a community context. The authors worked with community residents to identify an issue of concern to them and their children with disabilities: traffic safety. As you will see, specific roles for the investigators included developing a strong reciprocal partnership with the community, identifying issues the community cared about, helping design a coding system to observe the behavior in context, and developing action steps to address the issue in collaboration with the community. The community in this example refers to Latino parents of youth and young adults with disabilities.
Case Study 16.4
Familias Saludables (Healthy Families)
Yolanda Suarez-Balcazar and her team, in collaboration with residents of a predominantly Latino neighborhood and staff from a local community agency, initially designed a healthy lifestyles program called Healthy Families (Familias Saludables) to address a concern identified by the community about a lack of culturally relevant healthy lifestyles programming for their families with developmental disabilities (Suarez-Balcazar et al., 2016; Suarez-Balcazar et al., 2018). Eight evidence-based sessions were then provided that focused on enhancing healthy routines at home and in the community, and addressed neighborhood issues that families cared about. During these sessions, many mothers reported cars in their neighborhood driving over the speed limit or failing to stop for crosswalks, leading them to worry for their children’s safety and their own. Furthermore, one of her collaborators (The Consortium to Lower Obesity in Chicago Children, n.d.) was at the time sponsoring a city-wide effort to collect walkability data across Chicago’s diverse neighborhoods and address walkability safety concerns. The lack of pedestrian traffic safety in program participants’ neighborhoods further made worse the numerous environmental barriers that youth and young adults with Intellectual and Developmental Disabilities already faced to participating in their communities (e.g., over-stimulating sensory features, challenging physical layout). Utilizing Applied Behavior Analysis, the team designed a study in collaboration with their community partners to examine the behavior of traffic patterns in this neighborhood.
To assess the neighborhood’s walkability, data on traffic safety were collected based on direct observations. Independent researchers stood on street corners throughout the target area to observe and record traffic patterns including the number of cars failing to stop for the crosswalk, the smoothness of sidewalks, the speed of cars, and general street conditions. Data were collected at 25 intersections and focused on the behavior of car drivers on selected neighborhood streets, previously identified by community residents and agency staff as places/streets frequented by local families due to proximity to parks, schools, places of worship, and stores. Behaviors of interest that were recorded included drivers complying with street signs, drivers engaging in a complete stop whereby the car is not moving for a minimum of five seconds, and cars yielding to pedestrians.
They found that only one out of every three cars stopped at stop signs, and only half stopped before the crosswalk to allow pedestrians to pass. A pedestrian survey regarding walkability safety and engagement in the community found that participants walked to about six to nine places within the neighborhood weekly. Survey respondents also shared concerns about traffic safety. Consistent with a Behavioral Community Psychology approach, the data were utilized by the research team, the community residents, and staff from the local partner organization. For example, over fifteen families participated in a community health walk to raise awareness of safe driving practices. The health walk was organized by a number of organizations and participating families who decided to use this walk as an opportunity to educate the community about walkability safety. Parents and children created signs in both English and Spanish with self-selected messages such as, “Do not text and drive” and “Maneja con cuidado” [drive carefully] to carry during the walk. Then together, the team and the families joined the health walk alongside other community residents to continue promoting pedestrian safety in their areas.
This case study nicely illustrates the core values of Behavioral Community Psychology. First, the target population was actively involved in all aspects of the intervention, including identifying the concerns and taking action. Second, increased fostering of the target population’s sense of personal control and empowerment was accomplished by families designing the signs to carry during the health walk and identifying the streets to be observed. Third, there was the use of objective and subjective assessments when evaluating outcomes, using an instrument developed by a local traffic safety expert group. Fourth, they saw an increased emphasis on antecedent events, and in this case, an important antecedent was the traffic signs posted at street corners indicating to drivers that the desired behavior is for them to stop. Finally, increased interdisciplinary collaboration occurred, as this project included a team of community psychologists, several graduate students from occupational therapy and nutrition, and one faculty member from public health.
Some might think that focusing on dog litter or traffic safety is just not important, given the other types of serious problems that we face. But behavioral community psychologists have tackled even larger issues, as the next case study demonstrates.
Case Study 16.5
The I-Files: A Community Movement for the Big Island
The project began during the first day of a two-day behavioral training workshop in Honolulu in the spring of 1997. The workshop was on the development of community coalitions for health improvement and was delivered at the request of the Chronic Disease Branch Chief of the State of Hawai’i Department of Public Health. In attendance were many employees of the health departments from Honolulu and several of the neighboring islands. Also present were health educators, public health nurses, epidemiologists, branch chiefs from several divisions, and the Directors of District Health Offices from Kauai and Hilo. By noon on the first day, one table was absolutely buzzing with excitement. At the table were six public health nurses, three health educators, and the district health officer. On the back of a napkin, they sketched an idea for a community health improvement initiative that would later be named I-Files. The idea was simple—for each of the six districts on the Big Island, they would develop teams of three people including someone specializing in community mobilization and action planning, another person specializing in grant writing, and another person who would be trained in behavioral evaluation. They would develop community partnerships to address the most pressing health issues in each district, and mobilize the people and financial resources needed to resolve the issues.
Many of the behavioral programs and efforts that began in 1997 continue to this day. Some of the major behavioral accomplishments include: the development of a drug and alcohol-free surf competition for area teens; a community leadership development program where the fire department trained teens in firefighting, community service, and leadership skills; a community coalition comprised of businesses, professional fishermen, and community members in Kona to minimize destructive fishing practices near the area’s reefs. Further accomplishments were a community coalition to support the continued success of the Ka’u Hospital to promote community health improvement and build the capacity of residents to prevent health outcomes such as diabetes, the development of a neighborhood watch and diabetes self-management program by public health nurses, and a community policing program. To learn more about these outcomes, see the Prezi presentation .
In this case study, community members and employees of the District Health Offices were involved in a project that led to breakthrough approaches in community health improvement. By empowering the community health workers to respond to what they were learning from community members, more responsive programming was developed to meet real-world needs as they emerged in communities. By creating coalitions that represented multiple sectors in communities, programs and policies were developed that affected behavioral changes to reach more people, resulting in improvements in a variety of valued life outcomes. The new and modified programs, policies, and practices developed by and with community members represented a variety of behavior change strategies. By involving both the community members and the public health workforce, programs and policies were more relevant and had a long term positive influence. As this case example indicates, Behavioral Community Psychology investigators can make a difference using their innovative methods with a wide variety of community issues.
OTHER ISSUES TO CONSIDER
Clearly, the behavior of individuals and groups takes place in complex environments across a variety of settings including families, homes, schools, neighborhoods, community agencies, places of worship, and work settings. These behaviors can be complex as they are in constant interaction with and impacted by the environment.
Behavioral community interventions do not operate in a vacuum but rather compete with other competing messages, which can affect and influence the effectiveness of the interventions. As mentioned in Chapter 1, for many years smoking prevention interventions were implemented in schools, and yet children reported that they were almost always sold cigarettes by store vendors. The fact that youth were sold these dangerous substances did have a negative effect on school smoking prevention programs, for there were youth who wondered how dangerous cigarettes could be if they were sold cigarettes by adults. Therefore, additional behavioral interventions were needed to reduce youth access to retail sources of tobacco. As another example, commercials on the media constantly advertise the benefits of food that have high levels of sugar and fat, and these media influences will also need to be addressed by those working on childhood obesity behavioral programs.
Some feel there are no generalizable laws of behavior that can be used to understand something as complex as a community, which has complicated layers of ecological influences operating at the individual, group, organizational, and societal levels. In other words, an intervention that might work in one setting might not be effective in a different place, so they feel that behaviorists who believe in universal laws of behavior are overly simplistic. We have an answer to this concern. We believe that the principles of learning reviewed in this chapter are operating in community settings, but there might be important differences in how they are implemented in different settings (Jason et al., 2016). As an example, a behavioral school-based program that has been shown to work well in environments that have adequate staff and resources might have very different outcomes when located in an environment where there are inadequate resources and other multiple problems due to gangs and violence. What is important to remember is that an excellent intervention in one setting might not work well when it is implemented in another that is less supportive and reinforcing. Also, an intervention for one age group might now work for another age group, as indicated below in Case Study 16.6.
Case Study 16.6
Prevention of Prejudice
Behavioral Community psychologists attempted to impact inter-ethnic relations of elementary school age children by implementing a peer tutoring program. These youth were shown how to tutor their classmates. Direct observations of social interactions on the playground and sociometric indices of interpersonal ratings were used to measure inter-ethnic associations before and after the eight-week program. For first-graders, inter-ethnic interactions and sociometric choices increased and improvements were found in arithmetic and reading grades. However, no significant changes were found among the third-grade program children. The first-grade children carried the benefits of the peer tutoring program from the classroom to the playground. First-grade youngsters quickly developed cohesive, interdependent groups and some continued to use their group name from the study on the playground eight months following the program’s end. The findings suggest the importance of fostering cooperation and addressing prejudice early on, as indicated by the lack of significant changes for third-graders in their ratings of fellow children of different ethnic backgrounds. These findings further suggest that a behavioral cooperative peer tutoring classroom structure may improve the inter-ethnic relationships of first-grade children, who have experienced only a short history of competitive academic exercises and whose overt ethnic prejudice may be less ingrained (Rooney-Rebeck & Jason, 1986).
It is by attending to the unique history, age, and resources available in these school settings that we might be better about to understand these types of different outcomes. Basic behavioral principles are operating throughout these settings, and our task is to better identify and understand these complex environmental factors.
Some have also been concerned about the language and terms used by behavioral psychologists, who often used words such as “controlling behavior” (Mattaini, Jason, & Glenwick, 2016). With dictators using this type of repressive language to control their citizens, it is understandable that some might be concerned with the terms sometimes used by behaviorists. But these phrases do not capture the rich collaborations that occur in behavioral community interventions, when researchers work closely with community members in both defining the goals of interventions and actively participating in bringing about changes (Glenwick & Jason, 1980). As shown by the case studies above, behavioral community psychologists are able to gain control of key aspects of the environment, but it is through a collaborative, mutually agreed upon process with key members of the community (Bogat & Jason, 2000).
SUMMING UP
In this chapter, the principles of Applied Behavior Analysis were utilized in the Behavioral Community Psychology case studies to illustrate how to promote real community change in a variety of settings. Behavioral community psychologists focus on the complex relationship between individual/group behavior and the environment. By attending to the relationship between behavior and the environment, whereby each impacts the other, it is possible to bring about important change.
As we end this chapter on Behavioral Community Psychology, it is useful to point out that many of our social and community problems, such as substance abuse, school failure, and juvenile delinquency, share many developmental roots such as poverty and lack of resources. Coordinating our behavioral community interventions so that they uncover common environmental causes, and take them into consideration as we implement change, will ultimately increase the enduring impact of this work.
Critical Thought Questions
- Discuss what you think are some of the benefits and limitations of using Applied Behavior Analysis principles to study problems of interest to Community Psychology?
- Many community psychologists do research in schools. What do you think about the practice of some school teachers across the country using different types of tangible positive reinforcers (e.g., stickers, candy) to promote compliance and “good classroom behavior?” Justify your response.
- Imagine that you are living in a tenant building with an active renters’ association and you were elected president of the association for a one-year term. You noticed a problem with residents’ lack of awareness about the need to recycle. Although the recycling bins are provided and street collection is scheduled once per week, hardly anyone is taking the recycling bins out. Most tenants are young college students who are likely to produce recycling items (cans, paper, plastic). You want to start a recycling campaign and assess its effectiveness. Based on the principles learned in this chapter, identify components of the campaign and how you would get residents to recycle. Identify how you would assess if the campaign is effective. What would you measure as the unit of behavior for which you can take baseline measures?
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REFERENCES
Arellano, R., Balcazar, F. E., Alvarado, F., & Suarez, S. (2016). A participatory action research and intervention in a rural community of Mexico concerns report method/Titulo en Español. Universitas Psychologica , 15 (2), 15.
Balcazar, F., Garcia-Iriarte, E., & Suarez-Balcazar, Y. (2009). Participatory action research with Colombian immigrants. Hispanic Journal of Behavioral Sciences , 31 (1), 112-127.
Bandura, A. (1986). Social foundations of thought and action: A social cognitive theory . Prentice-Hall, Inc.
Bogat, G. A., & Jason, L. A. (2000). Toward an integration of behaviorism and community psychology. In J. Rappaport & E. Seidman (Eds.), Handbook of Community Psychology , (pp. 101-114). New York: Kluwer Academic/Plenum Publishers.
Clark, R. N., Burgess, R. L., & Hendee, J. C. (1972). The development of anti-litter behavior in a forest campground. Journal of Applied Behavior Analysis , 5 (1), 1-5.
Consortium to Lower Obesity in Chicago Children [CLOCC]. (n.d.). Neighborhood walkability initiative. Retrieved from http://www.clocc.net/our-focus-areas/physical-activity-and-built-environment/neighborhood-walkability-initiative/
Embry, D. D. (2002). The good behavior game: A best practice candidate as a universal behavioral vaccine. Clinical Child and Family Psychology Review , 5 (4), 273-297.
Fawcett, S. (1991). Some values guiding community research and action. Journal of Applied Behavior Analysis , 24 (4), 621-636.
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Jason, L. A., Glantsman, O., O’Brien, J. F., & Ramian, K. N. (2019). Introduction to the field of Community Psychology. In L. A. Jason, O. Glantsman, J. F. O’Brien, & K. N. Ramian (Eds.), Introduction to Community Psychology: Becoming an agent of change. Retrieved from https://press.rebus.community/introductiontocommunitypsychology/chapter/intro-to-community-psychology/
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Hanley, G., Iwata, B., & McCord, B. (2003). Functional analysis of problem behavior: A review. Journal of Applied Behavior Analysis , 36 (2), 147–185. Ka’u Rural Health Community Association. (n.d.). Home. Retrieved from https://krhcai.com/
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Mattaini, M. A., Jason, L. A., & Glenwick, D. S. (2016). Behavioral and time-series approaches. In L. A. Jason & D. S. Glenwick (Eds.), Handbook of methodological approaches to community-based research: Qualitative, quantitative, and mixed methods . (pp. 177-186). New York, NY: Oxford University Press.
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Suarez-Balcazar, Y., Early, A., Maldonado, A., Arias, D., Garcia, C., Zeidman, A., & Agudelo-Orozco, A. (2018). Community-based participatory research to promote healthy lifestyles among Latino immigrant families with youth with disabilities. Scandinavian Journal of Occupational Therapy .
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17.1B: Pericardium
The pericardium is a thick, membranous, fluid-filled sac which encloses, protects, and nourishes the heart.
- Distinguish between the fibrous and serous layers of the pericardium
Key Points
- The pericardium is a mesothelium tissue of the thoracic cavity which surrounds the heart. The outer layer, the fibrous pericardium, is comprised of dense connective tissue that protects the heart, anchors it to the surrounding walls, and prevents it from overfilling.
- The inner layer of the pericardium, the serous pericardium, is further divided into two layers, an outer parietal layer and an inner visceral layer with the pericardial cavity in between the two layers.
- The serous pericardium functions to lubricate the heart with pericardial fluid, which is stored in the space between the parietal and visceral layers.
- Pericarditis is inflammation of the pericardium, often due to infection, which can constrict and put harmful pressure on the heart.
Key Terms
- fibrous pericardium : Composed of dense connective tissue which protects the heart, anchors it to the surrounding walls, and prevents the heart from overfilling with blood.
- serous pericardium : Located deeper than the fibrous pericardium, this structure contains two layers that lubricate the heart to prevent friction from occurring during heart activity.
- pericardium : A serous membrane that surrounds and protects the heart.
The pericardium is the thick, membranous, fluid-filled sac that surrounds the heart and the roots of the vessels that enter and leave this vital organ, functioning as a protective membrane. The pericardium is one of the mesothelium tissues of the thoracic cavity, along with the pleura which cover the lungs. The pericardium is composed of two layers, an outer fibrous pericardium and an inner serous pericardium.
Membranes of the Thoracic Cavity : A transverse section of the thorax, showing the contents of the middle and the posterior mediastinum. The pleural and pericardial cavities are exaggerated since normally there is no space between parietal and visceral pleura and between pericardium and heart.
Fibrous Pericardium
The fibrous pericardium is the outer layer of the pericardium. It is composed of dense connective tissue which anchors the heart to the mediastinum of the chest wall. It prevents the heart from overfilling with blood and protects it from nearby infections by completely separating it from the rest of the thoracic cavity. It is continuous with the outer fibrous layer of the neighboring great blood vessels.
Serous Pericardium
The serous pericardium, the inner layer of the pericardium, is composed of two different layers. The outer layer, the parietal layer , is completely adhered to the fibrous pericardium. The inner layer is known as the visceral layer , which covers and protects the great vessels and heart. The space between the parietal and visceral layers is called the pericardial cavity.
The visceral layer is referred to as the epicardium in the areas where it is in direct contact with the heart. The space between these two serous layers, the parietal and the visceral, is the pericardial cavity, which contains pericardial fluid. The serous pericardium, with its two membranes and the fluid-filled pericardial cavity, provides protection to the heart and a lubricated sliding surface within which the heart can move in response to its own contractions and to the movement of adjacent structures such as the diaphragm and the lungs.
Function of the Pericardium
The pericardium is important because it protects the heart from trauma, shock, stress, and even infections from the nearby lungs. It supports the heart and anchors it to the medastinum so it doesn’t move within the body. The pericardium lubricates the heart and prevents it from becoming too large if blood volume is overloaded (though it will not prevent chronic heart enlargement).
Despite these functions, the pericardium is still vulnerable to problems of its own. Pericarditis is the term for inflammation in the pericardium, typically due to infection. Pericarditis is often a severe disease because it can constrict and apply pressure on the heart and work against its normal function. Pericarditis comes in many types depending on which tissue layer is infected.


5.8: Temporal Environment - The Daily Schedule
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One feature of a well-organized classroom is the use of a schedule and established routines. Schedules and established routines are important because they influence a child’s social and emotional development. While referred to as the daily schedule, it is important to recognize that the flow, or the predictable order of the day, should be the focus (rather than abiding by rigid timelines for the different parts of the day). While there may be parts of the day that at are at fixed times (for example meals or using a shared outdoor space), teachers should use flexibility to make the schedule meet the needs of the children. If an activity seems to be coming to a natural conclusion earlier, consider transitioning to the next part of the day. If children are really engaged in an activity, consider giving them additional time to wrap up their exploration. Flexibility also comes in handy when there are changes that affect the schedule that are beyond your control, such as bad weather preventing outdoor play. [1]
Schedules are important because they
Several factors influence child engagement.
The first component is “blocks of time,” the big chunks of time set aside for classroom activities. Preschool schedules typically include:
The next component is the sequence. Sequencing the blocks of time requires taking into consideration multiple factors including:
Schedules include some of the daily routines such as meal times, but may not include others such as bathroom breaks or clean up routines.
When planning the schedule you want to provide balance. This includes:
The daily schedule balances child-initiated play and teacher-guided activities. The latter involves teachers planning, introducing, and guiding specific activities to enhance children’s learning during small- and large-group times. In contrast, child-initiated play refers to children’s responses to ideas and materials introduced by teachers that the children are free to explore without teacher guidance. Child-initiated play also includes those times when children create, organize, and engage in activities completely on their own.
A daily schedule that ensures ample time for children to initiate their own play in well-developed interest areas is critical to the teaching and learning. Young children need ample time to engage in play, in the company of peers, in order to build their ideas, to pose problems, to try out solutions, and to negotiate and exchange ideas. When children initiate, organize, and develop their own play in the interest areas, it is called child-initiated learning. At times, children choose to play alone, but frequently, child-initiated play takes place in small groups of their own choosing.
In a schedule with ample time for children to initiate play in well-stocked interest areas, there are times when teachers organize and guide specific activities for children. Such teacher-guided curriculum activities are clearly distinct from child-initiated curriculum activities. Teacher-guided activities occur in two contexts—small groups and large groups. A small group would consist of one teacher working with a group of four to eight children. A large group is typically a gathering of all the children in an early childhood setting. Each context serves a different purpose and requires different preparation and different teaching strategies.
For some aspects of the curriculum, teachers may choose to organize an activity with a small group of children. Although initiated and guided by the teacher, an effective small-group encounter of this nature should still be rich in possibilities for children to contribute and negotiate ideas with each other. Teacher-guided activities in small groups work best in quiet spaces away from distractions of the full group and provide a manageable context for children to discuss and explore ideas and experiences. The teacher listens to children’s ideas, helps orchestrate the give-and-take of ideas among children, and poses ideas or problems for children to wonder about, explore together, or even solve. Away from the distractions of a large group, teachers can easily observe, listen, and converse with children in a small group, as well as note how individual children think, express ideas, relate with others, and use their emerging skills.
Such teacher-guided conversations can enrich children’s learning in all domains, particularly the children’s language and vocabulary development. In addition, teachers can intentionally guide the development of specific skills by planning small-group activities (e.g., songs, games, shared reading) for short periods of time that playfully engage children in using specific emerging skills.
Small-group activities have several advantages over large-group activities. With small groups of children, teachers can readily observe, listen, and document children’s developmental progress. Teachers can also individualize the curriculum and use questions or prompts to scaffold each child’s thinking in more complex ways.
Whether the activities are child-initiated or teacher-guided, children’s use of materials in interest areas provide teachers with excellent opportunities to observe how they build concepts and skills and how they negotiate ideas with others. Moments of observed play and interactions also provide teachers with ideas on how to extend children’s exploration and learning through future encounters with related materials that add novelty, challenge, and complexity in each domain.
Large groups provide another context for teacher-guided activities. The large group—typically a gathering of the entire class—works well for singing, acting out songs and stories, playing games, sharing experiences with each other, telling stories, building a sense of community, and organizing the daily schedule and activities. Storytelling is one of the more popular large-group experiences, one that has rich potential for adding to children’s understanding about the world around them. Storytelling allows teachers, children, family members, as well as storytellers from the community to tap into and build children’s knowledge and experiences in meaningful ways. Large-group time is also when teachers let the whole group of children know what new experiences will be available in the interest areas or what will happen in small groups that day. Large-group gatherings that occur at the end of the day provide opportunities to review noteworthy happenings and to anticipate what will be available the next day. [6]
Here are some examples of daily schedules for preschool classrooms.
|
Time of Day |
Routine/Activity |
Description |
|---|---|---|
|
8:00 |
Arrival/Greetings |
Wash Hands, Sign-in, Get Name tags |
|
8:05-8:20 |
Group Time |
Welcome, Songs, Stories, Discussions |
|
8:20-9:30 |
Open Choice Time Outdoors |
Explore classroom areas |
|
9:30 |
Clean-up |
Wash Hands, transition to indoors |
|
9:35-10:45 |
Snack/Open Choice Indoors |
|
|
10:45 |
Clean-up |
|
|
10:50-11 |
Closing Circle |
|
Time of Day |
Routine/Activity |
Description |
|---|---|---|
|
8:00 |
Arrival/Greetings |
Wash Hands, Sign-in, Get Name tags |
|
8:05-8:20 |
Group Time |
Welcome, Songs, Stories, Discussions |
|
8:20-9:30 |
Open Choice Time Outdoors |
Explore classroom areas |
|
9:30 |
Clean-up |
Wash Hands, transition to indoors |
|
9:35-11:00 |
Snack/Open Choice Indoors |
|
|
11:00 |
Clean-up |
|
|
11:10-11:25 |
Story Time |
Transition to wash hands for lunch |
|
11:30-11:50 |
Lunch |
|
|
11:50 |
Explore books on rest Mats |
|
|
12:10-2:00 |
Rest Time |
|
|
2:00-3:15 |
Indoor Choice Time/Snack |
|
|
3:15 |
Clean-up |
|
|
3:20 |
Large Group Circle |
Games, Songs, Stories |
|
3:40 |
Explore Outdoors |
|
|
5:00 |
Clean-up |
|
|
5:05 |
Indoor Small Group/Choice Centers |
Fewer areas open |
Schedules define the whole day, whereas routines are more specific sets of regularly occurring behaviors. Routines provide some security and a sense of what comes next; children are able to anticipate what will happen, and thus feel more secure.
Daily routines and rituals also provide a second context for curriculum. They offer possibilities for children to use their emerging skills and to apply emerging concepts and ideas. Early childhood daily routines include arrivals and departures, mealtimes, naptimes, diapering, toileting, dressing, handwashing, tooth-brushing, and transitions between one place and another. They also include rituals such as sign-in sheets, health checks, waiting lists, attendance counts, dictated stories, reminder notes, or voting.
Children sit down for a meal, wash their hands, and put jackets and shoes on hundreds of times in order to provide excellent opportunities for children to use and build emerging skills and concepts. In group care, the care routines during arrivals, departures, meals, naps, diapering, toileting, and dressing provide excellent opportunities for children to use and challenge their emerging skills and concepts. When an infant whose diaper is about to be changed hears her teacher describe what it is she is about to do, the infant experiences a flood of words, which eventually become an anticipated phrase that gives meaning to a familiar experience. When this same infant hears a request to put his arm into the sleeve of a shirt, he is invited to demonstrate that he has understood this phrase and experiences the joy that comes with sharing meaning with the teacher. When a preschool child looks in anticipation each morning at the helper chart to see what job she gets to do that day, they are invited not only to cooperate in the care of the classroom, but also to build their emerging skills in understanding the meaning of print that accompanies the photo or drawing. Care routines are natural opportunities for children to engage in learning. Therefore, teachers plan the routines of care and the daily rituals that pepper the day in ways that invite children to be active participants and to use and build their emerging skills and concepts in meaningful situations. [8]
A vignette featuring toddlers shows the kind of learning that occurs in another routine:
Vignette
Four toddlers are seated at a low table for lunch. Their primary care teacher sits with them at the table. To his right, on a low bench, the primary care teacher has a bin that holds everything he needs for the meal. He pulls out bibs for each toddler and helps each toddler put one on. Each toddler finds a cube chair to sit in. The teacher puts an empty bowl in front of the toddler on his left. He offers this toddler a pair of small plastic tongs, holds a plate of small sandwiches, and asks, “Would you like to take a sandwich?” The toddler grabs the tongs and, after a few trials, manages to pick up one of the sandwiches and drop it onto his plate. Later, after each toddler has taken a sandwich, the teacher pulls from the bin a clear plastic measuring cup, on which a red line is drawn at the one-cup mark. He fills the measuring cup to the red line. He places an empty glass in front of a toddler and, offering the toddler the measuring cup, says, “Would you like to pour?” The toddler wraps his hand around the handle and tips the cup over his glass. He spills a bit at first, but adjusts his hand and manages to empty the measuring cup. He looks up at the teacher and smiles. The teacher smiles in response, saying, “You poured your milk, Stephan! You know how to do it!” The toddler seated next to Stephan reaches for the empty measuring cup. The teacher says, “And now you can pour milk into your glass, Alexi. I’ll put the milk in the measuring cup first.” [9]
[10]
It is important to focus on creating and managing smooth transitions between activities in the classroom. Reasons to address transitions between activities in early childhood classrooms include:
There are numerous strategies that can be used to ensure well-organized transitions between activities. These include strategies you use before the transition, during the transition, and following the transition.
You can also work to promote independence during transitions by
The following vignette offers an opportunity to watch and listen for the learning that occurs during a transition routine and to reflect on the planning that had to occur in order for this experience to play out as it did.
Vignette
Ms. Cone had used the children’s name tags in transition activities for quite some time, at first pointing out and naming the first letter in each name as she called children to go wash hands or to get their jackets before going outside. Somewhat later, she held up each of the nametags and pointed to the first letter as she asked the child to name it. Today, she is using the first sounds in names to send a few children at a time from the circle time area to wash hands for lunch: “If your name starts with /k/, you may go wash your hands. Yes, Connie and Carolina, you may go to the sink. Both of your names start with the /k/ sound.” Cindy sees Connie and Carolina stand up, and she stands up too. Ms. Cone explains that Cindy begins with the /s/, not /k/ sound, and that she’ll get a turn soon. Cindy says, “I’m a C too!” Ms. Cone says, “Oh, you are right. Your name begins with the letter c like Connie and Carolina, but it starts with a different sound. We hear /k/ at the beginning of Connie and Carolina—/k/ Connie, /k/ Carolina. We hear /s/ at the beginning of your name—/s/—Cindy. I’m going to say that sound next: ‘If your name starts with /s/, you may go wash your hands.’” Sabrina stood up, joined hands with Cindy, and they walked to the sink together. [12]
Built into this large-group gathering is a dismissal ritual that takes full advantage of young children’s interest in their names and the names of their friends. As part of this dismissal ritual, the teacher invites children to use their emerging skills in distinguishing the distinct sounds of language, described in the language and literacy foundations as phonological awareness. She embeds this learning in the context of a game, one that inspires children to listen carefully to the sounds spoken in instructions for inviting small groups of children to wash hands. The transition from large group to the sink area goes much more smoothly as a result, and in the process, children get to use an important emerging skill.
[1] Guide to Managing the Classrooms: Schedules and Routines by the U.S. Department of Health and Human Services is in the public domain;
Content by Jennifer Paris is licensed under CC BY 4.0
[2] Guide to Managing the Classrooms: Schedules and Routines by the U.S. Department of Health and Human Services is in the public domain
[3] Image by the U.S. Department of Health and Human Services is in the public domain
[4] Image by the U.S. Department of Health and Human Services is in the public domain
[5] Guide to Managing the Classrooms: Schedules and Routines by the U.S. Department of Health and Human Services is in the public domain
[6] California Preschool Curriculum Framework, Volume 3 by the California Department of Education is used with permission
[7] Based on the College of the Canyons Early Childhood Education Sample schedule
[8] Guide to Managing the Classrooms: Schedules and Routines by the U.S. Department of Health and Human Services is in the public domain;
The Integrated Nature of Learning by the California Department of Education is used with permission (pg. 29-32)
[9] The Integrated Nature of Learning by the California Department of Education is used with permission
[10] Tips for Teachers by the U.S. Department of Health and Human Services is in the public domain
[11] Guide to Presenting Managing the Classroom: Classroom Transitions by the U.S. Department of Health and Human Services is in the public domain
[12] The Integrated Nature of Learning by the California Department of Education is used with permission


26.3: Anti-derivatives and integrals
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In the previous section, we were concerned with determining the derivative of a function \(f(x)\) . The derivative is useful because it tells us how the function \(f(x)\) varies as a function of \(x\) . In physics, we often know how a function varies, but we do not know the actual function. In other words, we often have the opposite problem: we are given the derivative of a function, and wish to determine the actual function. For this case, we will limit our discussion to functions of a single independent variable.
Suppose that we are given a function \(f(x)\) and we know that this is the derivative of some other function, \(F(x)\) , which we do not know. We call \(F(x)\) the anti-derivative of \(f(x)\) . The anti-derivative of a function \(f(x)\) , written \(F(x)\) , thus satisfies the property:
\[\begin{aligned} \frac{dF}{dx}=f(x)\end{aligned}\]
Since we have a symbol for indicating that we take the derivative with respect to \(x\) ( \(\frac{d}{dx}\) ), we also have a symbol, \(\int dx\) , for indicating that we take the anti-derivative with respect to \(x\) :
\[\begin{aligned} \int f(x) dx &= F(x) \\[4pt] \therefore \frac{d}{dx}\left(\int f(x) dx\right) &= \frac{dF}{dx}=f(x)\end{aligned}\]
Earlier, we justified the symbol for the derivative by pointing out that it is like \(\frac{\Delta f}{\Delta x}\) but for the case when \(\Delta x\to 0\) . Similarly, we will justify the anti-derivative sign, \(\int f(x) dx\) , by showing that it is related to a sum of \(f(x)\Delta x\) , in the limit \(\Delta x\to 0\) . The \(\int\) sign looks like an “S” for sum.
While it is possible to exactly determine the derivative of a function \(f(x)\) , the anti-derivative can only be determined up to a constant. Consider for example a different function, \(\tilde F(x)=F(x)+C\) , where \(C\) is a constant. The derivative of \(\tilde F(x)\) with respect to \(x\) is given by:
\[\begin{aligned} \frac{d\tilde{F}}{dx}&=\frac{d}{dx}\left(F(x)+C\right)\\[4pt] &=\frac{dF}{dx}+\frac{dC}{dx}\\[4pt] &=\frac{dF}{dx}+0\\[4pt] &=f(x)\end{aligned}\]
Hence, the function \(\tilde F(x)=F(x)+C\) is also an anti-derivative of \(f(x)\) . The constant \(C\) can often be determined using additional information (sometimes called “initial conditions”). Recall the function, \(f(x)=x^2\) , shown in Figure A2.2.1 (left panel). If you imagine shifting the whole function up or down, the derivative would not change. In other words, if the origin of the axes were not drawn on the left panel, you would still be able to determine the derivative of the function (how steep it is). Adding a constant, \(C\) , to a function is exactly the same as shifting the function up or down, which does not change its derivative. Thus, when you know the derivative, you cannot know the value of \(C\) , unless you are also told that the function must go through a specific point (a so-called initial condition).
In order to determine the derivative of a function, we used Equation A2.2.1 . We now need to derive an equivalent prescription for determining the anti-derivative. Suppose that we have the two pieces of information required to determine \(F(x)\) completely, namely:
- the function \(f(x)=\frac{dF}{dx}\) (its derivative).
- the condition that \(F(x)\) must pass through a specific point, \(F(x_0)=F_0\) .
The procedure for determining the anti-derivative \(F(x)\) is illustrated above in Figure A2.3.1. We start by drawing the point that we know the function \(F(x)\) must go through, \((x_0,F_0)\) . We then choose a value of \(\Delta x\) and use the derivative, \(f(x)\) , to calculate \(\Delta F_0\) , the amount by which \(F(x)\) changes when \(x\) changes by \(\Delta x\) . Using the derivative \(f(x)\) evaluated at \(x_0\) , we have:
\[\begin{aligned} \frac{\Delta F_0}{\Delta x} &\approx f(x_0)\;\;\;\; (\text{in the limit} \Delta x\to 0 )\\[4pt] \therefore \Delta F_0 &= f(x_0) \Delta x\end{aligned}\]
We can then estimate the value of the function \(F_1=F(x_1)\) at the next point, \(x_1=x_0+\Delta x\) , as illustrated by the black arrow in Figure A2.3.1
\[\begin{aligned} F_1&=F(x_1)\\[4pt] &=F(x+\Delta x) \\[4pt] &\approx F_0 + \Delta F_0\\[4pt] &\approx F_0+f(x_0)\Delta x\end{aligned}\]
Now that we have determined the value of the function \(F(x)\) at \(x=x_1\) , we can repeat the procedure to determine the value of the function \(F(x)\) at the next point, \(x_2=x_1+\Delta x\) . Again, we use the derivative evaluated at \(x_1\) , \(f(x_1)\) , to determine \(\Delta F_1\) , and add that to \(F_1\) to get \(F_2=F(x_2)\) , as illustrated by the grey arrow in Figure A2.3.1:
\[\begin{aligned} F_2&=F(x_1+\Delta x) \\[4pt] &\approx F_1+\Delta F_1\\[4pt] &\approx F_1+f(x_1)\Delta x\\[4pt] &\approx F_0+f(x_0)\Delta x+f(x_1)\Delta x\end{aligned}\]
Using the summation notation, we can generalize the result and write the function \(F(x)\) evaluated at any point, \(x_N=x_0+N\Delta x\) :
\[\begin{aligned} F(x_N) \approx F_0+\sum_{i=1}^{i=N} f(x_{i-1}) \Delta x\end{aligned}\]
The result above will become exactly correct in the limit \(\Delta x\to 0\) :
\[F(x_N) = F(x_0)+\lim_{\Delta x\to 0}\sum_{i=1}^{i=N} f(x_{i-1}) \Delta x\]
Let us take a closer look at the sum. Each term in the sum is of the form \(f(x_{i-1})\Delta x\) , and is illustrated in Figure A2.3.2 for the same case as in Figure A2.3.1 (that is, Figure A2.3.2 shows \(f(x)\) that we know, and Figure A2.3.1 shows \(F(x)\) that we are trying to find).
As you can see, each term in the sum corresponds to the area of a rectangle between the function \(f(x)\) and the \(x\) axis (with a piece missing). In the limit where \(\Delta x\to 0\) , the missing pieces (shown by the hashed areas in Figure A2.3.2) will vanish and \(f(x_i)\Delta x\) will become exactly the area between \(f(x)\) and the \(x\) axis over a length \(\Delta x\) . The sum of the rectangular areas will thus approach the area between \(f(x)\) and the \(x\) axis between \(x_0\) and \(x_N\) :
\[\begin{aligned} \lim_{\Delta x\to 0}\sum_{i=1}^{i=N} f(x_{i-1}) \Delta x=\text{Area between f(x) and x axis from $x_0$ to $x_N$}\end{aligned}\]
Re-arranging Equation A2.3.1 gives us a prescription for determining the anti-derivative:
\[\begin{aligned} F(x_N) - F(x_0)&=\lim_{\Delta x\to 0}\sum_{i=1}^{i=N} f(x_{i-1}) \Delta x\end{aligned}\]
We see that if we determine the area between \(f(x)\) and the \(x\) axis from \(x_0\) to \(x_N\) , we can obtain the difference between the anti-derivative at two points, \(F(x_N)-F(x_0)\)
The difference between the anti-derivative, \(F(x)\) , evaluated at two different values of \(x\) is called the integral of \(f(x)\) and has the following notation:
\[\int_{x_0}^{x_N}f(x) dx=F(x_N) - F(x_0)=\lim_{\Delta x\to 0}\sum_{i=1}^{i=N} f(x_{i-1}) \Delta x\]
As you can see, the integral has labels that specify the range over which we calculate the area between \(f(x)\) and the \(x\) axis. A common notation to express the difference \(F(x_N) - F(x_0)\) is to use brackets:
\[\begin{aligned} \int_{x_0}^{x_N}f(x) dx=F(x_N) - F(x_0) =\big [ F(x) \big]_{x_0}^{x_N}\end{aligned}\]
Recall that we wrote the anti-derivative with the same \(\int\) symbol earlier:
\[\begin{aligned} \int f(x) dx = F(x)\end{aligned}\]
The symbol \(\int f(x) dx\) without the limits is called the indefinite integral . You can also see that when you take the (definite) integral (i.e. the difference between \(F(x)\) evaluated at two points), any constant that is added to \(F(x)\) will cancel. Physical quantities are always based on definite integrals, so when we write the constant \(C\) it is primarily for completeness and to emphasize that we have an indefinite integral.
As an example, let us determine the integral of \(f(x)=2x\) between \(x=1\) and \(x=4\) , as well as the indefinite integral of \(f(x)\) , which is the case that we illustrated in Figures A2.3.1 and A2.3.2 . Using Equation A2.3.2 , we have:
\[\begin{aligned} \int_{x_0}^{x_N}f(x) dx&=\lim_{\Delta x\to 0}\sum_{i=1}^{i=N} f(x_{i-1}) \Delta x \\[4pt] &=\lim_{\Delta x\to 0}\sum_{i=1}^{i=N} 2x_{i-1} \Delta x \end{aligned}\]
where we have:
\[\begin{aligned} x_0 &=1 \\[4pt] x_N &=4 \\[4pt] \Delta x &= \frac{x_N-x_0}{N}\end{aligned}\]
Note that \(N\) is the number of times we have \(\Delta x\) in the interval between \(x_0\) and \(x_N\) . Thus, taking the limit of \(\Delta x\to 0\) is the same as taking the limit \(N\to\infty\) . Let us illustrate the sum for the case where \(N=3\) , and thus when \(\Delta x=1\) , corresponding to the illustration in Figure A2.3.2:
\[\begin{aligned} \sum_{i=1}^{i=N=3} 2x_{i-1} \Delta x &=2x_0\Delta x+2x_1\Delta x+2x_2\Delta x\\[4pt] &=2\Delta x (x_0+x_1+x_2) \\[4pt] &=2 \frac{x_3-x_0}{N}(x_0+x_1+x_2) \\[4pt] &=2 \frac{(4)-(1)}{(3)}(1+2+3) \\[4pt] &=12\end{aligned}\]
where in the second line, we noticed that we could factor out the \(2\Delta x\) because it appears in each term. Since we only used 4 points, this is a pretty coarse approximation of the integral, and we expect it to be an underestimate (as the missing area represented by the hashed lines in Figure A2.3.2 is quite large).
If we repeat this for a larger value of N, \(N=6\) ( \(\Delta x = 0.5\) ), we should obtain a more accurate answer:
\[\begin{aligned} \sum_{i=1}^{i=6} 2x_{i-1} \Delta x &=2 \frac{x_6-x_0}{N}(x_0+x_1+x_2+x_3+x_4+x_5)\\[4pt] &=2\frac{4-1}{6} (1+1.5+2+2.5+3+3.5)\\[4pt] &=13.5\end{aligned}\]
Writing this out again for the general case so that we can take the limit \(N\to\infty\) , and factoring out the \(2\Delta x\) :
\[\begin{aligned} \sum_{i=1}^{i=N} 2x_{i-1} \Delta x &=2 \Delta x\sum_{i=1}^{i=N}x_{i-1}\\[4pt] &=2 \frac{x_N-x_0}{N}\sum_{i=1}^{i=N}x_{i-1}\end{aligned}\]
Now, consider the combination:
\[\begin{aligned} \frac{1}{N}\sum_{i=1}^{i=N}x_{i-1}\end{aligned}\]
that appears above. This corresponds to the arithmetic average of the values from \(x_0\) to \(x_{N-1}\) (sum the values and divide by the number of values). In the limit where \(N\to \infty\) , then the value \(x_{N-1}\approx x_N\) . The average value of \(x\) in the interval between \(x_0\) and \(x_N\) is simply given by the value of \(x\) at the midpoint of the interval:
\[\begin{aligned} \lim_{N\to\infty}\frac{1}{N}\sum_{i=1}^{i=N}x_{i-1}=\frac{1}{2}(x_N+x_0)\end{aligned}\]
Putting everything together:
\[\begin{aligned} \lim_{N\to\infty}\sum_{i=1}^{i=N} 2x_{i-1} \Delta x &=2 (x_N+x_0)\lim_{N\to\infty}\frac{1}{N}\sum_{i=1}^{i=N}x_{i-1}\\[4pt] &=2 (x_N-x_0)\frac{1}{2}(x_N+x_0)\\[4pt] &=x_N^2 - x_0^2\\[4pt] &=(4)^2 - (1)^2 = 15\end{aligned}\]
where in the last line, we substituted in the values of \(x_0=1\) and \(x_N=4\) . Writing this as the integral:
\[\begin{aligned} \int_{x_0}^{x_N}2x dx=F(x_N) - F(x_0)=x_N^2 - x_0^2\end{aligned}\]
we can immediately identify the anti-derivative and the indefinite integral:
\[\begin{aligned} F(x) &= x^2 +C \\[4pt] \int 2xdx&=x^2 +C\end{aligned}\]
This is of course the result that we expected, and we can check our answer by taking the derivative of \(F(x)\) :
\[\begin{aligned} \frac{dF}{dx}=\frac{d}{dx}(x^2+C) = 2x\end{aligned}\]
We have thus confirmed that \(F(x)=x^2+C\) is the anti-derivative of \(f(x)=2x\) .
Exercise \(\PageIndex{1}\)
The quantity \(\int_{a}^{b}f(t)dt\) is equal to
- the area between the function \(f(t)\) and the \(f\) axis between \(t=a\) and \(t=b\)
- the sum of \(f(t)\Delta t\) in the limit \(\Delta t\to 0\) between \(t=a\) and \(t=b\)
- the difference \(f(b) - f(a)\) .
- Answer
-
Common anti-derivative and properties
Table A2.3.1 below gives the anti-derivatives (indefinite integrals) for common functions. In all cases, \(x,\) is the independent variable, and all other variables should be thought of as constants:
| Function, \(f(x)\) | Anti-derivative, \(F(x)\) |
|---|---|
| \(f(x)=a\) | \(F(x)=ax+C\) |
| \(f(x)=x^n\) | \(F(x)=\frac{1}{n+1}x^{n+1}+C\) |
| \(f(x)=\frac{1}{x}\) | \(F(x)=\ln(|x|)+C\) |
| \(f(x)=\sin(x)\) | \(F(x)=-\cos(x)+C\) |
| \(f(x)=\cos(x)\) | \(F(x)=\sin(x)+C\) |
| \(f(x)=\tan(x)\) | \(F(x)=-\ln(|\cos(x)|)+C\) |
| \(f(x)=e^x\) | \(F(x)=e^x+C\) |
| \(f(x)=\ln(x)\) | \(F(x)=x\ln(x)-x+C\) |
Table A2.3.1 : Common indefinite integrals of functions.
Note that, in general, it is much more difficult to obtain the anti-derivative of a function than it is to take its derivative. A few common properties to help evaluate indefinite integrals are shown in Table A2.3.2 below.
| Anti-derivative | Equivalent anti-derivative |
|---|---|
| \(\int (f(x)+g(x)) dx\) | \(\int f(x)dx+\int g(x) dx\) (sum) |
| \(\int (f(x)-g(x)) dx\) | \(\int f(x)dx-\int g(x) dx\) (subtraction) |
| \(\int af(x) dx\) | \(a\int f(x)dx\) (multiplication by constant) |
| \(\int f'(x)g(x) dx\) | \(f(x)g(x)-\int f(x)g'(x) dx\) (integration by parts) |
Table A2.3.2 : Some properties of indefinite integrals.
Common uses of integrals in Physics - from a sum to an integral
Integrals are extremely useful in physics because they are related to sums. If we assume that our mathematician friends (or computers) can determine anti-derivatives for us, using integrals is not that complicated.
The key idea in physics is that integrals are a tool to easily performing sums . As we saw above, integrals correspond to the area underneath a curve, which is found by summing the (different) areas of an infinite number of infinitely small rectangles. In physics, it is often the case that we need to take the sum of an infinite number of small things that keep varying, just as the areas of the rectangles.
Consider, for example, a rod of length, \(L\) , and total mass \(M\) , as shown in Figure A2.3.3. If the rod is uniform in density, then if we cut it into, say, two equal pieces, those two pieces will weigh the same. We can define a “linear mass density”, \(\mu\) , for the rod, as the mass per unit length of the rod:
\[\begin{aligned} \mu = \frac{M}{L}\end{aligned}\]
The linear mass density has dimensions of mass over length and can be used to find the mass of any length of rod. For example, if the rod has a mass of \(M=5\text{kg}\) and a length of \(L=2\text{m}\) , then the mass density is:
\[\begin{aligned} \mu=\frac{M}{L}=\frac{(5\text{kg})}{(2\text{m})}=2.5\text{kg/m}\end{aligned}\]
Knowing the mass density, we can now easily find the mass, \(m\) , of a piece of rod that has a length of, say, \(l=10\text{cm}\) . Using the mass density, the mass of the \(10\text{cm}\) rod is given by:
\[\begin{aligned} m=\mu l=(2.5\text{kg/m})(0.1\text{m})=0.25\text{kg}\end{aligned}\]
Now suppose that we have a rod of length \(L\) that is not uniform, as in Figure A2.3.3, and that does not have a constant linear mass density. Perhaps the rod gets wider and wider, or it has a holes in it that make it not uniform. Imagine that the mass density of the rod is instead given by a function, \(\mu(x)\) , that depends on the position along the rod, where \(x\) is the distance measured from one side of the rod.
Now, we cannot simply determine the mass of the rod by multiplying \(\mu(x)\) and \(L\) , since we do not know which value of \(x\) to use. In fact, we have to use all of the values of \(x\) , between \(x=0\) and \(x=L\) .
The strategy is to divide the rod up into \(N\) pieces of length \(\Delta x\) . If we label our pieces of rod with an index \(i\) , we can say that the piece that is at position \(x_i\) has a tiny mass, \(\Delta m_i\) .   The total mass, \(M\) , of the rod is then the sum of the masses of the tiny rods, in the limit where \(\Delta x\to 0\) :
\[\begin{aligned} M &= \lim_{\Delta x\to 0}\sum_{i=1}^{i=N}\Delta m_i \\[4pt] &= \lim_{\Delta x\to 0}\sum_{i=1}^{i=N} \mu(x_i) \Delta x\end{aligned}\]
But this is precisely the definition of the integral ( Equation A2.3.1 ), which we can easily evaluate with an anti-derivative:
\[\begin{aligned} M &=\lim_{\Delta x\to 0}\sum_{i=1}^{i=N} \mu(x_i) \Delta x \\[4pt] &= \int_0^L \mu(x) dx \\[4pt] &= G(L) - G(0)\end{aligned}\]
where \(G(x)\) is the anti-derivative of \(\mu(x)\) .
Suppose that the mass density is given by the function:
\[\begin{aligned} \mu(x)=ax^3\end{aligned}\]
with anti-derivative ( Table A2.3.1 ):
\[\begin{aligned} G(x)=a\frac{1}{4}x^4 + C\end{aligned}\]
Let \(a=5\text{kg/m}^{4}\) and let’s say that the length of the rod is \(L=0.5\text{m}\) . The total mass of the rod is then:
\[\begin{aligned} M&=\int_0^L \mu(x) dx \\[4pt] &=\int_0^L ax^3 dx \\[4pt] &= G(L)-G(0)\\[4pt] &=\left[ a\frac{1}{4}L^4 \right] - \left[ a\frac{1}{4}0^4 \right]\\[4pt] &=5\text{kg/m}^{4}\frac{1}{4}(0.5\text{m})^4 \\[4pt] &=78\text{g}\\[4pt]\end{aligned}\]
With a little practice, you can solve this type of problem without writing out the sum explicitly. Picture an infinitesimal piece of the rod of length \(dx\) at position \(x\) . It will have an infinitesimal mass, \(dm\) , given by:
\[\begin{aligned} dm = \mu(x) dx\end{aligned}\]
The total mass of the rod is the then the sum (i.e. the integral) of the mass elements
\[\begin{aligned} M = \int dm\end{aligned}\]
and we really can think of the \(\int\) sign as a sum, when the things being summed are infinitesimally small. In the above equation, we still have not specified the range in \(x\) over which we want to take the sum; that is, we need some sort of index for the mass elements to make this a meaningful definite integral. Since we already know how to express \(dm\) in terms of \(dx\) , we can substitute our expression for \(dm\) using one with \(dx\) :
\[\begin{aligned} M = \int dm = \int_0^L \mu(x) dx\end{aligned}\]
where we have made the integral definite by specifying the range over which to sum, since we can use \(x\) to “label” the mass elements.
One should note that coming up with the above integral is physics. Solving it is math. We will worry much more about writing out the integral than evaluating its value. Evaluating the integral can always be done by a mathematician friend or a computer, but determining which integral to write down is the physicist’s job!