Skip to lesson

Choose “Save as PDF” in the print dialog to keep a copy.

Cogito

AP Calculus AB · Chapter 8 · Lesson 2

Practice Worksheet

Volumes of Revolution

Spinning a region into a solid.

12 problems · about 22 minutes · CHA-5.C

NameDate

Figure — use these to answer the problems

A stack of slices makes a volume. A solid of revolution stacks circular slices instead of rectangular ones.
  • Cubes in one layer4 × 3 = 12
  • Layers2
  • 4 × 3 × 224 cubic units
Length4
Width3
Height2
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. Outer radius 4, inner radius 3. Washer area divided by π?

    Answer

  2. What is the most common error in the washer method?

    • Squaring the difference of the radii instead of subtracting the squares.
    • Using the wrong limits.

  3. A disk of radius 4. Area divided by π?

    Answer

  4. Outer radius 6, inner radius 2. Washer area divided by π?

    Answer

  5. Is π(R² − r²) the same as π(R − r)²? 1 yes, 0 no.

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. Rotating about the x-axis, which variable do you integrate in? 1 x, 2 y.

    Answer

  2. Rotating about the y-axis, which variable? 1 x, 2 y.

    Answer

  3. Rotating y = f(x) about the line y = 2. The radius is f(x) minus what?

    Answer

Stretch Yourself

Mixed problems. Work out what kind of question it is before you start.

4 problems

  1. Sort each situation by which method it needs.

    Write each item under the heading it belongs to: The region touches the axis of rotation · Solid slices with no hole · The region is bounded away from the axis · Slices with a hole in the middle

    Disk method

    Washer method

  2. Outer radius 10, inner radius 6. Washer area divided by π?

    Answer

  3. The Disk

    A disk has radius 3. What is its area divided by π?

    Answer

  4. The Washer

    Outer radius 5, inner radius 3. What is the washer area divided by π?

    Answer