After learning what integrals are and how to evaluate them, students build on what they know from Geometry and apply their understanding of integrals to find the volume of a special class of solids called solids of revolution. This can be a tricky application for students, but by understanding exactly which points of the process are difficult and why, we can use targeted strategies to help students confidently approach these types of problems. Here are our five tips for helping students better understand, set-up, and solve volume problems.
1. Visualize it.
What is a solid of revolution?
Students need to be able to visualize a 2-d region being rotated around a line to produce a 3-d solid of revolution. I like to use honeycomb tissue paper decorations to demonstrate this. Start by showing students the flat version, held up to a line drawn on the whiteboard . Ask students, “what do you think would happen if we rotated this shape around this line?” After students guess at the shape, actually open the decoration to show students the 3-d solid and its symmetry. Use a variety of decorations in various sizes. Spheres are the most common but bells, ice cream cones, or fruit help students consider more interesting 2-d regions and the shapes they make when rotated around an axis of revolution.
2. Conceptualize it.
Finding volume by slicing into cylinders
To help students move toward finding the volume of such a shape, we have students find the volume of a pear by cutting it into slices (Lesson 8.9). It is easy to see that each of these slices becomes a cylinder and not all slices have the same radius. Students see that the radius of each slice depends on where we are on the pear. This observation is critical to understanding why we must use an algebraic expression to denote the radius of a solid of revolution, because the radius is changing, and dependent on where the slice occurs in the region (at what x-value). By the end of this lesson, students should understand why the volume formula when using disks is
and why the radius will be an expression in terms of x or y and not a constant. More practice will have to be done to determine the exact expression for the radius (See Tip 4!).
We use bagels in Lesson 8.11 to visualize slices that are in the shapes of washers and students understand why and how the hollow center of the slice must be subtracted from the volume of the cylindrical slice they calculated already. By the end of this lesson, students should understand why the volume formula when using washers is
which can be written with a single integral as
Although the second form is more common, the long form ties more closely with their process for finding volume and helps them avoid the common error of using the square of the difference of the two radii instead of the difference of the squares of the two radii in their integrand expression.
3. Essentialize it.
Horizontal and vertical distances in the coordinate plane
Students need to be able to write a variable expression in terms of x or y that represents the radius of each slice. This is probably the hardest part for students and I believe the challenge lies in an incomplete understanding of a big mathematical idea from earlier grades: distance is a difference. Finding the length of a radius is finding the distance between two points that share an x-coordinate or a y-coordinate. Finding the change in the two coordinates, or the difference between them, requires subtracting two coordinates. In the scenario of radii, we are only looking for positive lengths, so we will always consider the absolute difference between the coordinates.
To solidify students’ understanding of distance in the coordinate plane, I spend a couple minutes showing students a variety of horizontal and vertical segments on the coordinate plane, asking them to find their length and asking them how they know they are correct. The first couple segments are solely in one quadrant, but the rest span two quadrants. Students must understand that the length of a segment between (-3,5) and (7,5) is 10 because the change in coordinates is 7-(-3)=10. Finally, I ask my students to find the length of the purple, green, and black segments in the graph below.
Instead of being given the two endpoints, students realize they must first find the y-value by evaluating the function at the specific x-value and then finding the distance between that y-value and y=-2. After doing this calculation 3 times, students see the pattern of taking the function output and then adding 2 (or subtracting negative 2). This is a chance for students to look for and express regularity in repeated reasoning (Mathematical Practice 8). You may want to repeat this exercise when the segments are horizontal and the curve is defined in terms of y.
While this exercise may feel trivial for a Calculus class, I have found it to be the single most helpful tool in helping students successfully set up their volume integrals. By identifying the essential mathematics behind a complex procedure, we can help students build strong conceptual understanding that will allow them to flexibly solve a variety of problems.
4. Connect it.
Writing expressions for the radii
Now that students had a refresher on how to find lengths in the coordinate plane, it’s time to connect this understanding to volume problems. When students see a volume problem, I always encourage them to draw sample radii from the axis of revolution to the upper boundary of the region (big radius) and to the lower boundary of the region (small radius) if they are using washers, and a single sample radius if they are using disks. The exercise explained above helps them clearly define an expression for each of these radii.
The phrases “upper - lower” and “further right - closer” can also help students remember how to set up an expression for the length of a radius if the radius is vertical or if the radius is horizontal, respectively.
When students understand conceptually that they are adding up infinitely many thin cylindrical disks or washers, and can fluently write expressions for the radius, the majority of the question is completed. While students still need to evaluate the integral (usually with the help of a calculator), the kinds of errors that occur here tend to be more mechanical than conceptual.
5. Practice it.
Solving volume of solids of revolution problems
Students need plenty of practice with finding volumes of solids of revolution. They should be able to quickly identify whether to use disks or washers, and whether their integrand should be written in terms of x or in terms of y. We suggest using a variety of released Free Response Questions from previous AP Calculus exams.