Cogito
AP Calculus AB · Chapter 8 · Lesson 2
Volumes of Revolution
Spinning a region into a solid.
12 problems · about 22 minutes · CHA-5.C
What this lesson teaches
The student computes volumes of solids of revolution using disks and washers.
- Rotating a region sweeps out a solid, sliced into disks or washers.
- A disk has area πr²; a washer has π(R² − r²).
- Square the radii first, then subtract.
Warm Up
Straightforward practice. Get the method working first.
5 problemsOuter radius 4, inner radius 3. Washer area divided by π?
Answer 7
Why 7.
What is the most common error in the washer method?
Answer Squaring the difference of the radii instead of subtracting the squares.
Why Squaring the difference.
A disk of radius 4. Area divided by π?
Answer 16
Why 4².
Outer radius 6, inner radius 2. Washer area divided by π?
Answer 32
Why 36 − 4.
Is π(R² − r²) the same as π(R − r)²? 1 yes, 0 no.
Answer 0
Why Square first, then subtract.
Build It Up
The same ideas with more to keep track of.
3 problemsRotating about the x-axis, which variable do you integrate in? 1 x, 2 y.
Answer 1
Why Slice perpendicular to the axis.
Rotating about the y-axis, which variable? 1 x, 2 y.
Answer 2
Why Slice perpendicular to that axis.
Rotating y = f(x) about the line y = 2. The radius is f(x) minus what?
Answer 2
Why The distance to the axis.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsSort each situation by which method it needs.
Answer Disk method: The region touches the axis of rotation, Solid slices with no hole · Washer method: The region is bounded away from the axis, Slices with a hole in the middle
Why A gap from the axis creates the hole.
Outer radius 10, inner radius 6. Washer area divided by π?
Answer 64
Why 100 − 36.
The Disk: A disk has radius 3. What is its area divided by π?
Answer 9
Why 9.
The Washer: Outer radius 5, inner radius 3. What is the washer area divided by π?
Answer 16
Why 16.