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

The region M is enclosed by the functions

*f*(*x*) =*x*^{2}+ 1 and*g*(*x*) = 2*x*+ 1. 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. Homework Help ✎The

*x*-axis.The

*y*-axis.The line

*y*= 5.The line

*x*= 4.

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

Use shells.

Note the *x* represents the length of the radius, it's part of the base = 2π*r* equation. And *f*(*x*) represents the height of each shell.

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 *f*(*x*) and *g*(*x*), since the axis of rotation is 5 units above the *x*-axis. Adjust the integral in part (a) accordingly:

As in part (b), to avoid using horizontal rectangles (and rewriting the integrandin terms of *y*), 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 *k* units to the left or right: