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

The region and

*. Set up, but do not evaluate, the integrals to represent the volumes of the solids formed by revolving*

The

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

The

-axis.Use shells.

Note the

represents the length of the radius, it's part of theequation. And represents the height of each shell.

The line

.Refer to the hint in part (a).

Recall that

andrepresent radii of an outer and inner solid (the 'inner' solid is the hole). Makethe function with the longer radii andthe function with the shorter radii. Note: It is possible thatis belowon the graph. Once rotation happens, this will switch.Also note that both radii have different lengths than

and, since the axis of rotation isunits above the -axis. Adjust the integral in part (a) accordingly:

The line

.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: