Since you are rotating around the

axis, this will be a

integral.

The graphs intersect where the equations are equal...

or

.

When

, so they intersect at

.

The volume is calculated by rotating the area around the

axis.

This area needs to be thought of as a series of rectangles, which when rotated, become cylinders.

The length of each rectangle is

and the width is

, a small change in

.

When rotated, each cylinder has a radius the same as the length of each rectangle, so the circular cross-sectional area is

. The height of each cylinder is the same as the width of each rectangle,

.

Therefore the volume of each cylinder is

.

So the entire volume can be approximated by

, and when you increase the number of cylinders, this sum converges on an integral and the approximation becomes exact.

So

.