Washer method

Volume of revolution with a hole: $V=\pi\int_a^b\left(\left[R(x)\right]^2-\left[r(x)\right]^2\right)dx$ .

The washer method extends the disk method to a region revolved about an axis that leaves a HOLE down the middle: each cross-section is an annulus (washer shape) with outer radius $R(x)$ and inner radius $r(x)$ , giving $V=\pi\int_a^b\left(\left[R(x)\right]^2-\left[r(x)\right]^2\right)dx$ — the outer disk's volume minus the inner disk's.

Examples

Revolving the region between $y=x$ and $y=x^2$ about the $x$ -axis produces a washer with outer radius $x$ and inner radius $x^2$ .