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

Round your answer to the nearest thousandth.

We first express the curve in a form that makes it easy to use the disk method. The region is bounded by

Since the curve is given as , we solve for to express the boundary in terms of :

When we revolve the region about the -axis, for a fixed the horizontal slice runs from to . This slice, when revolved about the -axis, forms a disk (since the inner radius is zero) with

The area of one such disk is

The -values range from the bottom of the region up to the top where the curve meets the -axis. When , the curve gives

Thus, the limits for are and .

The volume of the solid is then given by

We next expand the integrand:

So the volume integral becomes

Now, we integrate term by term:

Putting everything together:

To combine these terms, we use a common denominator, which is :

Thus,

So, the volume is

Now, rounding to the nearest thousandth, we approximate:

Thus, the volume of the solid is