§5.2 Homework Problems
Directions: For each problem, you will need to find the volume of the solid generated by revolving the
region bounded by the given curves about the specified line. To do this, complete the following steps,
as outlined in class:
1 Sketch the region and the solid on separate sets of axes. Label each curve appropriately.
2
a. Will your representative slice will be vertical or horizontal? Will your slab be a disk or a
washer?
b. Draw a representative slice in the 2d region, and sketch a typical disk or washer in the 3d
solid. Label the appropriate widths, heights and radii.
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) Do only step 4 for y = x and y = x2 ; about y = 2.
π
2. y = x, y = x2 ; about the y-axis. Ans:
6
π
3. y = x2 , x = 1, y = 0; about the x-axis. Ans:
5
2π
4. x + 2y = 2, x = 0, y = 0; about the x-axis. Ans:
3
2
5. y = x , 0 ≤ x ≤ 2, y = 4, x = 0; about the y-axis. Ans: 8π
16π
6. x = y 2 , x = 1; about x = 1. Ans:
15
√
π
7. y = x, y = x; about y = 1. Ans:
6
√
π
8. y = x, y = x; about x = 1. Ans:
5