Find the volume of the solid
formed by revolving the region
bound by the graphs ,
, and about the
-axis. Use the disk method.

Round your answer to the nearest thousandth.

We first note that the region is bounded by

For any in , the -coordinate of the curve is

which is negative in this interval (since and thus ). The region between and is located below the -axis. However, when we revolve the region about the -axis, the vertical distance between and becomes the radius of a typical disk. That is, for a fixed , the radius is

Using the disk method, the volume is given by

Since the integrand is an even function, we can simplify the integral by doubling the integral from 0 to 2:

First, expand the integrand:

Thus,

Now, compute the antiderivative term by term:

Substitute these into the formula:

Thus, the volume is

To express the volume in a decimal approximation, we have

Rounded to the nearest thousandth, the volume is approximately