8.9 Volume with Disc Method: Revolving Around the x- or y-Axis

Setting up the disc method

When a region is revolved around the x-axis or y-axis, each cross-section perpendicular to the axis is a circular disc. The disc's radius equals the distance from the axis to the curve. Volume = π ∫ [R(x)]² dx (revolving about x-axis) or π ∫ [R(y)]² dy (about y-axis), integrated over the interval bounding the region.

Choosing the right variable

Always integrate with respect to the variable matching the axis of revolution. Revolving around the x-axis means slicing vertically, radius = f(x), and integrating in dx. Revolving around the y-axis means slicing horizontally, radius = g(y), and integrating in dy. If the function isn't already solved for the needed variable, solve for it first.

Radius from the axis, not just the function

The radius is the distance from the axis of revolution to the curve, not necessarily the function value itself. If the axis is shifted (e.g., y = k instead of y = 0), the radius becomes |f(x) − k|. Always sketch the region to correctly identify bounds and radius.

Key terms

Disc method formula (about x-axis)

V = π ∫ from a to b of [f(x)]² dx

Disc method formula (about y-axis)

V = π ∫ from c to d of [g(y)]² dy

How to choose integration variable

Match the axis of revolution: x-axis → integrate dx; y-axis → integrate dy.

Radius when axis is shifted (e.g., y = k)

Radius = |f(x) − k|, the distance from the curve to the shifted axis.

Shape of cross-section in disc method

A solid circular disc with no hole; area = πr².

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. 4π
4. 8