§5.3 Homework Problems
Directions: For each problem, find the volume of the solid generated by revolving the region bounded
by the given curves about the specified line using the method of shells, unless specified otherwise. To
do this, complete the following steps, as outlined in class:
1 Sketch the region and the solid on the same set of axes.
2 Draw a representative slice and cylindrical shell. Label the appropriate width, height and radius.
3 Approximate the volume of the solid Vn and set up the integral V = lim Vn .
n→∞
4 Integrate to find V .
The answers appear next to (most) problems to help you.
1. (from class) Use the washer method for the problem we completed in class, and compare your
answer to that from the method of shells: y = x2 , y = 1, x = 0; about the x-axis.
π
2. y = x(x − 1)2 , y = 0; about the y-axis. Ans:
15
21π
2
3. x = 1 + y , x = 0, y = 1, y = 2; about the x-axis. Ans:
2
2
2
4. y = x − 6x + 10, y = −x + 6x − 6; about the y-axis. Ans: 16π
√
64π
5. x = y, x = y 3 /32; about the x-axis. Ans:
5
29π
2
2
6. y = x , x = y ; about y = −1. Ans:
30
256π
7. y = 4x − x2 , y = 8x − 2x2 ; about x = −2. Ans:
3
√
208π
8. x = y, x = y 3 /32; about y = 4. Ans:
15
67π
9. y = x2 , y = 0, x = 1, x = 2; about x = 4. Ans:
6