8.10 Disc Method (around other axes)

Shifted axis

When revolving around a line other than a coordinate axis (like y = 2 or x = −1), the radius is the distance from the curve to that line: radius = |f(x) − (line)|.

Same formula

The volume is still V = π∫[radius]² dx — only the radius expression changes to account for the shifted axis.

Key terms

Shifted-axis disc

Radius = distance from the curve to the line.

Around y = k

Radius = |f(x) − k|.

Formula

Still V = π∫[radius]² dx.

What changes

Only the radius expression.

Radius meaning

Distance to the axis of revolution.

Try an AP-style question

The region bounded by y = x², y = 1, and x ≥ 0 is revolved about y = 1. What is the volume?

1. 4π/15
2. 8π/15
3. 2π/3
4. π/5