Integrating Areas to Get Volumes

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SWBAT explain and solve problems involving integrating cross-sectional areas of solids to compute their volumes.

Big Idea

Integrating area gives volume. But why? Students will answer this question in two different but related ways.

Video Narrative - Lesson Overview

Warm-Up + Homework Review

30 minutes

For today’s Warm-Up I ask students to distinguish between total distance and displacement in a thoughtfully chosen velocity function to illustrate the difference.  A velocity function that changes sign at some point, or equivalently crossing through the x-axis, will demonstrate to students the distinction between velocity and speed. 

If my students do not bring it up in our discussion of the warm-up, I will question students about the relative sizes of the total distance and the displacement values. I want to focus their thinking on when these two quantities are equal and when they are not.  I expect my students to notice the negative area under the velocity function reflecting over the x-axis to represent positive area under the speed function, which justifies why total distance is always greater than or equal to displacement and why the total distance cannot be negative.  

Teacher’s Note:  Last night’s homework solutions appear in the In The Classroom file.

Note that on part E of last night’s homework, the total distance and displacement are equal (for t > 0) since  v(t) = 4 + 8/(t+ 1) is always positive.  But, students should learn to always ask this question just-in-case the car changes direction on the interval.


Setting the Stage

50 minutes

If necessary, I will take some time today to wrap up the Variable Limits FTC Exploration worksheet from yesterday.  Additionally, I often find it worthwhile to spend some extra time with the Trace Area Under Curve v1 applet.  I typically ask students to describe whether the traced integral function is increasing/decreasing and concave up/down, and to identify local extrema and inflection points based on the graphical behavior of the first derivative function.   

Prior to today's review of volumes of solids (later in today’s lesson), it will be important for my students to master finding areas between curves.  Students remember the mnemonic “Big Y minus Little Y” very well, but often forget where it comes from or why it works.  Recreating the Areas Between Curves graphic helps students visualize subtracting the red area between g(x) and the x-axis from the green area between f(x) and the x-axis.  Akin to cutting cookies out of cookie dough, what remains is the area we wanted to find.

I find my students have two common misconceptions about areas between curves. 

  1. Students sometimes switch which function represents Big Y and which represents Little Y.  Another way you might phrase this mnemonic is “Top Y” and “Bottom Y”, to emphasize which function has the greater y-values.  This might be especially confusing if both functions are below the x-axis, in which case the function farthest from the x-axis is actually the Little Y since it’s y-values are more negative, whereas students might think that function should be Big Y since it’s y-values are farther away from the x-axis.
  2. The second misconception is that the location of the x-axis matters.  I challenge my students to explain why the area of the region would change if the x-axis was farther above/below the region bounded by the two functions, or equivalently, if both functions were translated up or down by the same amount.  

I share additional ideas for pursuing these misconceptions in Differentiating this Lesson.

After completing this discussion, the In The Classroom file contains released AP multiple choice questions related to finding areas between curves.   I use PollEverywhere (see my Polleverywhere In The Classroom video) for students to submit their answers via text to promote active engagement among all students.


35 minutes

I open this chunk of the lesson by introducing the Big Idea. We are transitioning from calculating areas to determining volumes of solids.  I find that students have good intuitions about how this might happen: Integration by “reversing” the power rule for derivatives means the exponent increases by one.  Informally then, it makes sense that integrating an area expression measured in square units will result in cubed units, which corresponds to volume as is familiar to students. 

I also discuss a concrete example. We can think about slicing a loaf of bread very thinly, so thinly that it only makes sense to calculate the area of each slice, and then summing the areas of all of the infinitely thin slices to “reassemble” the loaf of bread and determine its volume.  In this way, we can extend our work with finding areas between curves into finding volumes of solids when those areas are revolved around a line or when those areas represent cross-sectional slices of a solid.

Volumes of Solids of Revolution
For solids of revolution, the In The Classroom file defines a region and asks students to revolve this region around horizontal lines above, below, and on a boundary of the region, and vertical lines to the left, right, and on a boundary of the region.  If students still struggle with setting up the correct integrals then I will first give time for students to discuss the problems with each other without beginning to actually solve each problem on their papers, and after ample time direct students back to working individually to solve each problem.

This format preserves the opportunity for students to get guidance from each other when it is needed, but also attends to the need to solve these problems individually in a few weeks on the AP exam and our final exam.  Through repetition, students should recognize the similar structure (SMP #7) that exists in setting up the disk method for the lines on a boundary of the region and the washer method for the lines not on a boundary of the region and thereby leaving a “donut hole” when revolved around the other lines.  Some students experience difficulty with algebraic aspects of this lesson (see Watching for Misconceptions).

To wrap up this chunk of the lesson, I remind students of their work on the Bottle Project for plotting a region from the edge of a bottle then revolving that region around a line to get its volume.

Cross-sectional Volumes of Solids
Two-at-a-time, I display the shapes at the end of the In The Classroom file.  I ask my students to work individually to calculate the area of each shape using their knowledge of basic geometry.  The figures on the left give a numeric value for the length of one of the sides, but the figures on the right give that same side defined in terms of a difference between two functions.  The purpose here is for students to fluently operate with quantities whether they are constants or variables. Ultimately, I want students to be able to integrate the variable expressions for Area representing cross-sectional slices of a solid to obtain its volume. At this point in the year, having learned this content just a few months ago, I expect my students should be comfortable transitioning from the numeric to the variable case.

If my students struggle to make this transition, however, I will likely ask them to solve several more cases with constant side lengths. I will, however, give the figures wacky measurements like 5.0799318 and tell students to carry through this decimal without simplifying along the way.  If students can do this, then carrying through a variable is no different than carrying through the wacky decimal. 

To wrap up this chunk of the lesson, I remind students of their in-class project using pipe cleaners to construct models of volumetric solids with a variety of cross-sectional shapes.


Teacher's Note: The next two lessons will continue working the problems from today's In The Classroom file, so there is no need to rush through these problems or attempt them all in today’s lesson.  Students will need time to recall and apply their prior learning.  If students need a mini-reteaching on a particular topic, I will take the time in class to provide this adapted instruction, and pick up wherever I leave off in the next class.



Closure + Homework

5 minutes

If I choose to go for a less active but fascinating closure, I might play one of many videos on YouTube animating volumes of solids.  Here is one of them, but there are many others you might prefer over this one. 

For a more traditional closure, I will give students a minute to think independently and then explain to a neighbor how

  1. A loaf of bread
  2. The 2nd and 3rd dimensions

can be used to explain why integrating area gives volume. 

I tell each pair that one neighbor gets to explain the bread approach. The other explains the dimensions approach.

Here is tonight's homework: Integrating Areas Homework.docx