8.11 Volume with Washer Method: Revolving Around the x- or y-Axis

When to use washers

Use the washer method when revolving a region around an axis creates a solid with a hole — meaning the region is bounded between two curves rather than a curve and the axis. Each cross-section is a washer (a disk with a smaller disk removed) instead of a solid disk.

Setting up the integral

For revolution around the x-axis: V = π∫[a,b] (R(x)² − r(x)²) dx, where R(x) is the outer radius (farther curve) and r(x) is the inner radius (closer curve). For the y-axis, integrate with respect to y using R(y) and r(y), swapping roles of x and y accordingly.

Finding radii correctly

Radii are measured as distance from the axis of revolution to each curve, not the curve values themselves. Always identify which curve is outer vs. inner over the interval, and find intersection points to determine integration bounds. Sketching the region first prevents sign and setup errors.

Key terms

Washer method (x-axis)

V = π∫[a,b] (R(x)² − r(x)²) dx, where R is the outer radius and r is the inner radius.

When to use washer vs. disk

Use washer when the region does not touch the axis of revolution, creating a hole in the solid.

Outer radius R(x)

The distance from the axis of revolution to the curve farther away.

Inner radius r(x)

The distance from the axis of revolution to the curve closer to it.

Finding integration bounds

Set the two boundary curves equal to each other and solve for x (or y) to find intersection points, which become the limits a and b.

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/6
2. π/2
3. π/15
4. 2π/15