8.9 Disc Method (around an axis)

Solids of revolution

Revolving a region around an axis with no gap produces discs. Around the x-axis: V = π∫[a to b] [f(x)]² dx (radius = f(x)). Around the y-axis, integrate in y.

Each disc

Each disc has area π·(radius)². Use discs when the region touches the axis (no hole).

Key terms

Disc method

Volume of a solid of revolution with no gap.

Around x-axis

V = π∫ [f(x)]² dx.

Disc area

π·(radius)².

When to use

Region touches the axis (no hole).

Around y-axis

Integrate in y.

Try an AP-style question

The region under y = √x on [0, 4] is revolved about the x-axis. What is the volume?

1. 16π
2. 8π
3. 8
4. 4π