8.11 Washer Method (around an axis)

When there's a gap

When there is a gap between the region and the axis, revolving produces washers (discs with a hole). V = π∫ ([R_outer]² − [R_inner]²) dx.

Two radii

The outer radius reaches the curve farther from the axis; the inner radius reaches the nearer curve. Subtract the inner-radius area from the outer.

Key terms

Washer method

For a region with a gap from the axis.

Formula

V = π∫ (R_outer² − R_inner²) dx.

Outer radius

To the curve farther from the axis.

Inner radius

To the curve nearer the axis.

Washer

A disc with a hole in the middle.

Try an AP-style question

The region between y = x and y = x² is revolved about the x-axis. What is the volume?

1. π/2
2. π/15
3. 2π/15
4. π/6