### Home > CALC > Chapter 8 > Lesson 8.4.1 > Problem 8-142

The region M is enclosed by the functions and

*. Set-up, but do not evaluate, the integrals to represent the volumes of the solids formed by revolving M about each of the following axes.*

The

-axis.Sketch the axis of rotation (flag pole) then use washers.

The

-axis.Use shells.

Note therepresents the length of the radius, it's part of the baseequation. And represents the height of each shell.

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.Refer to the hint in part (a).

Recall that

*R*and*r*represent radii of an outer and inner solid (the 'inner' solid is the hole). Make*R*the function with the longer radii and*r*the function with the shorter radii. Note: It is possible that*R*is below*r*on the graph. Once rotation happens, this will switch.Also note that both radii have different lengths than

andsince the axis of rotation is, units above the -axis. Adjust the integral in part (a) accordingly:

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.As in part (b), to avoid using horizontal rectangles (and rewriting the integrand in terms of

), you can use shells.The setup should look exactly like the setup in part (b), with one exception: The radii are no longer the same values as the bounds. They will be either longer or shorter than the bounds. So shift the radii

units to the left or right: