# Solutions to Problem Set 12

Math 211-03

10-2-2017

[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 x-axis.

Use circular slices. The radius of a typical slice is . The thickness of a typical slice is . The volume is

2. Let R be the area bounded by and the x-axis, from to . Find the volume of the solid generated when R is revolved about the x-axis.

Use circular slices. The radius of a typical slice is . The thickness of a typical slice is . The volume is

3. Let R be the area in the first quadrant bounded above by and below by . Find the volume of the solid generated when R is revolved about the y-axis.

Use circular slices. The radius of a typical slice is . The thickness of a typical slice is . The volume is

4. Let R be the region between and the x-axis, from to . Find the volume generated by revolving R about the x-axis.

Use circular slices. The radius of a typical slice is . The thickness of a typical slice is . The volume is

5. Let R be the region between and , from to . Find the volume of the solid generated when R is revolved about the x-axis.

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

6. 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

He who walks in another's tracks leaves no footprints. - Joan Brannon

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