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
Today’s activity transitions students from solid slices (disks) to the concept of volumes by washers. By slicing a bagel and creating several washers, most students will be able to intuit the volume formula used in the table on page 2.
Because the term washer may need to be defined for some students, clearly differentiate between the two types of slices used to find volumes of revolution: a washer is simply a solid disk with a hole in the center. To help students visualize the stacking of washers used in the table, encourage them to sketch on their graph one or two slices of the “bagel” formed by revolving their circle around the x-axis. Use intentional vocabulary when describing the orientation of disks and slices. Consider describing the slices of the bagel as “vertical washers” or “dx washers” to clearly indicate that the thickness (width) of the washer will be measured as a change in x. This terminology will also plant the seed that a “dx” integral is required to find their bagel volume.
Integral expressions for volumes by washers are complicated and complex, but most errors can be eliminated by an accurate sketch and a consistent approach to solving. Ask the same questions in the same order or consider using a solving template (see Day 12).
The practice problems found in CYU consider rotations around a major axis: either the x-axis or the y-axis. To prepare students for other axes of revolution, we have introduced subtraction of zero into the integrands. This small move not only reinforces our terminology of “upper minus lower” or “further minus closer” when creating integrand expressions, but better prepares students to generalize to other axes of revolution in the coming lessons.
Questions involving volume by washers most often appear on the free response section of the AP exam. Because the given functions for the outer and inner radii are often complicated, students may be asked only to set up a volume integral instead of finding a numeric value. This allows readers to award points for correct integrand expressions, limits of integration, or the inclusion of constants (namely, pi).
Continue to remind students that a vertical radius (for slices or washers) indicates a dx integral with limits of integration moving from left to right: the thickness of the washer is being measured along the x-axis. Likewise, a horizontal radius indicates a dy integral because the limits of integration travel from bottom to top as we measure the thickness of these horizontal washers on the y-axis.