# Solutions to Problem Set 13

Math 211-03

10-3-2017

[Volumes of revolution]

1. Let R be the region in the first quadrant cut off by the line . Find the volume of the solid generated when R is revolved about the line .

Use washers. The inner radius of a typical washer is .

The outer radius of a typical washer is

The thickness of a typical washer is . The volume is

2. Let R be the region inside the square defined by the inequalities and .

Find the volume generated when R is revolved about the y-axis.

The inner radius is and the outer radius is . The thickness of a typical washer is . The volume is

3. Let R be the region in the first quadrant cut off by the line . Find the volume of the solid generated when R is revolved about the line .

Use cylindrical shells. The radius of a typical shell is , the height of a typical shell is , and the thickness is . The volume is

Note: It's possible to do this problem using washers.

4. Let R be the region in the first quadrant cut off by the line . Find the volume of the solid generated when R is revolved about the line .

Use cylindrical shells. The radius of a typical shell is , the height of a typical shell is , and the thickness is . The volume is

Note: It's possible to do this problem using washers.

5. Let R be the region bounded by the graph of and the x-axis, from to . Find the volume generated by revolving R about the y-axis.

It's possible to do this problem using washers, but it would be messier. Instead, use cylindrical shells. The radius of a typical shell is , the height of a typical shell is , and the thickness is . The volume is

Here's the work for the integral, which is done by parts:

6. Let R be the region bounded by the curves and . Find the volume generated by revolving R about the line .

Find the intersection points of the curves:

Use cylindrical shells.The radius of a typical shell is , the height of a typical shell is

The thickness is . The volume is

Note: You could do this problem using washers, but it would require 2 integrals. Do you see why?

We'd be wise to question why we hold a grudge as if it were going to make us happy and ease our pain. It's rather like eating rat poison and thinking the rat will die. - Pema Chödrön

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