Calculate volumes of solids of revolution using definite integrals
I can determine when solids generated by revolution will have hollow spaces
I can identify both radii of the washer by using “upper-lower” or “further right-closer”
I can set up a definite integral to find volume using the washer method
Quick Lesson Plan
In this activity, students work in groups of four on whiteboards to solve four distinct, but related problems. After each group member has completed their problem, all four answers will be summed and the final answer presented to the teacher. If the sum is incorrect, group members must work collaboratively to find each others’ errors.
To prepare for this activity, we suggest printing each page of the document on different colored cardstock. You may wish to make multiple copies of each problem set as groups will finish at various times and will need a new problem to work on. Arrange students in groups of four. Have students assign each group member a letter A-D which they will stick with for the whole activity.
Hand out one card per team (order doesn’t matter). Using a large whiteboard, each group member will work on their part of the problem (Player A does part a, Player B does part b, etc.). The problem sets are specifically designed so that each student will see a variety of problem types (washers/disks, in terms of x/y, and revolving around the major axes and other axes). When all group members are finished, have students sum up their four answers and call you over to check. If the sum is correct, hand them a new problem set. If the sum is incorrect, group members must work collaboratively to find each others’ errors. Do not tell them which of the four responses are correct or incorrect. Suggest to students that they turn the whiteboard 90 degrees so they can look at the work of another group member or discuss as a group which parts are likely to contain errors.This activity promotes excellent discussion among students and critical thinking about how to set-up the integrals to find volume of solids of revolution, as well as how to input these expressions into their calculator. In our experience, working on whiteboards allows students to clearly see each other’s work and thus enhances discussion and accountability.
One modification is to write the correct sum on the middle of the whiteboard when you hand out the problem set, allowing students to continue working until they arrive at the correct sum, without having to call the teacher over.
Remind students to label intersection points on their paper so they don’t have to keep writing all 6 decimal points. Also remind students how to store the boundary functions on their calculator as Y1 and Y2, in order to make less errors with parentheses.
In most years, there is an FRQ dedicated to area and volume. Students could be asked to find the volume of a solid of revolution with respect to a horizontal or vertical axis, using the washer or the disk method. These questions can be calculator or non-calculator, so make sure students know how to use the appropriate calculator keys as well as how to integrate by hand. Volume using cross sections is also often a part of these FRQs, and will be introduced in subsequent days.
As students are in the early learning phase of volume of solids of revolution consider offering the Problem Solving Template as a scaffold. This document provides students with a sequence of questions that they must consider when approaching a volume problem, as well as an organizational structure that will help them break down a complex process. Some students may not need this scaffold, and others may only need it on the first day. Others may find it a helpful tool for several days, but be sure that the scaffold is removed later in the learning process.
Writing an expression for the big and small radius when revolving around another axis is a challenge for students. Reinforce the idea of “upper-lower” and “further right-closer”. Build conceptual understanding by asking students the distance between x=-3 and x=4 and having students make explicit how they found this distance.
Another common mistake is to square the difference of the two radii instead of subtracting the squares of the two radii. Help students conceptualize the shape of a washer as a small circle cut out of a larger circle.