Volumes of Revolution Practice Problems with Answers

These volumes of revolution practice problems cover the disk, washer and shell methods, rotation around the y-axis, and rotation around lines that are not axes. Try each one first. The answers come first so you can check quickly, then the full solutions.

The three formulas, for rotation around the x-axis or the y-axis:

For pictures of each slice, read the volumes of revolution guide.

Problems

Disks around the x-axis

1. y = \sqrt{x}, 0 \le x \le 4, around the x-axis
2. y = x^2, 0 \le x \le 1, around the x-axis
3. y = \sqrt{9 - x^2}, -3 \le x \le 3, around the x-axis
4. y = \sin x, 0 \le x \le \pi, around the x-axis

Washers

5. The region between y = x and y = x^2, around the x-axis
6. The region between y = \sqrt{x} and y = x, around the x-axis

Around the y-axis

7. The region between y = x^2, y = 4 and the y-axis (x \ge 0), around the y-axis, with disks
8. The region under y = x - x^2 on [0, 1], around the y-axis, with shells
9. The region under y = x^2 on [0, 2], around the y-axis, with shells

Other axes

10. The region between y = x and y = x^2, around the line y = -1
11. The region between y = x^2 and y = 1, around the line y = 1

Answers

Step-by-step solutions

1. Each slice is a disk of radius \sqrt{x}:

2. The radius is x^2, so R^2 = x^4:

3. The radius squared is 9 - x^2:

This is a sphere of radius 3, and \frac{4}{3}\pi(3)^3 = 36\pi. Good check.

4. Use \sin^2 x = \frac{1 - \cos 2x}{2}:

5. On [0, 1], the line y = x is farther from the axis, so R = x and r = x^2:

6. On [0, 1], \sqrt{x} \ge x, so R = \sqrt{x} and r = x:

7. Slice horizontally. At height y, the radius is x = \sqrt{y}, so R^2 = y:

8. A shell at x has radius x and height x - x^2:

9. A shell at x has radius x and height x^2:

Problems 7 and 9 describe different solids that happen to have the same volume. Problem 7 is the solid that fills the paraboloid bowl; problem 9 is the solid under the bowl, between the bowl and the cylinder x = 2. Together they fill a cylinder of radius 2 and height 4, with volume 16\pi.

10. Measure from y = -1. The outer radius is x + 1 and the inner radius is x^2 + 1:

11. The region touches the line y = 1, so use disks. The radius is 1 - x^2, and the curves meet at x = \pm 1:

Common mistakes on these problems

• Writing (R - r)^2. In problem 5, \pi\int(x - x^2)^2\,dx gives \frac{\pi}{30}, not \frac{2\pi}{15}. Square each radius separately.
• Measuring from the wrong axis. In problem 10, the radii are x + 1 and x^2 + 1, not x and x^2.
• Forgetting the \pi or the 2\pi. The disk and washer formulas start with \pi. The shell formula starts with 2\pi.
• Mixing variables. Horizontal slices need everything in y, including the limits.

Keep practicing

For unlimited solids with instant feedback, try the disks lesson, the washers lesson and the cross sections lesson. If a region runs off to infinity, the improper integral problems show how to handle it.