8.12 Washer Method (around other axes)

Shifted axis with a gap

Revolving a region that has a gap around a line other than an axis (like y = −1) uses the washer method — but both radii are measured as distances from the curves to that line.

Same formula

V = π∫ (R_outer² − R_inner²) dx. Shifting the axis changes both the inner and outer radii.

Key terms

Shifted-axis washer

Region with a gap revolved around a non-axis line.

Both radii

Distances from the curves to the line.

Formula

V = π∫ (R_outer² − R_inner²) dx.

Outer radius

Distance to the farther curve.

What changes

Both radii shift with the axis.

Try an AP-style question

The region between y = x and y = x² is revolved about y = −1. What is the volume?

1. 2π/15
2. 7π/15
3. π/3
4. 8π/15