The Washer Method
The disc method can be extended to cover solids of revolution with a hole by replacing the representative disc with a representative washer. The washer is formed by revolving a rectangle about an axis, seen below. If r and R are the inner and outer radii of the washer and w is the width of the washer, then the volume is given by V = pi(R2 - r2)w.

Washer 1

To find the volume of a solid of revolution created by a washer, use the formula below.

The Washer Method
             b
V = pi §  ([R(x)]2 - [r(x)]2) dx.
             a

Note that the integral involving the inner radius represents the volume of the hole and is subtracted from the integral involving the outer radius.

Wash Method Wash Method Animation
Click here for an example problem.