Calculate volumes of solids of revolution using definite integrals
I can visualize solids generated by revolution around various axes
I can identify the radius of the disk by using “upper-lower” or “further right-closer”
I can set up a definite integral to find volume using the disc method
Quick Lesson Plan
This is a high-energy, high-engagement lesson to introduce the slices (disk) method for volumes. Be prepared for plenty of valuable conversation within groups as students create and manipulate the slices. Groups discover that the radius for each slice is actually the value of the function, which reappears in the integral formula for volumes by slices.
The lesson can be adapted for lemons, limes, and oranges (less interesting), and perhaps coconuts if you have appropriate tools on hand. We used both pears and lemons and our results were exceptional in each mode.
Students should be able to “predict” the ending of this lesson: we will be transitioning from n slices to an infinite number of slices by allowing the width (height of the cylinders) to approach zero: our delta x becomes the dx term. Use summation notation for a fixed number of cylinders; introduce a limit as the number of cylinders approaches infinity, then transition to integral notation as a reminder of the original development of integrals.
CYU #1 and #2 illustrate standard rotations around the x- and y-axes; CYU #3 can be accomplished with two integrals or one integral and the formula for the volume of a cone.
One of the most valuable aspects of this lesson is independent student learning. The “aha” moments occurred spontaneously as students had a physical representation of their work.
Exam Insight: Although volumes by slices are generally a “friendly” event for students, this technique presents a challenge to students when the problem is presented in context or the variables are not the traditional Cartesian pairs. 2016 AB5 definitely fit both categories. The “funnel problem” should be practiced by every AP Calculus student before the exam.