Volumes of Revolution — Mathematics Knowledge System

Overview

Volumes of revolution are solids formed by rotating a two-dimensional region around an axis. Their volumes are computed by integrating cross-sectional areas (disk/washer methods) or cylindrical shell surface areas (shell method) along the axis of rotation.

Intuition

If you rotate a region around an axis, slicing the resulting solid perpendicular to that axis produces circular (disk) or ring-shaped (washer) cross-sections whose areas you can compute directly and then sum via an integral. Alternatively, slicing parallel to the axis produces thin cylindrical shells, each with easily computed circumference × height × thickness — the shell method.

Interactive Graph

Formal Definition

Definition

The three standard formulas for computing volumes of revolution:

V = \pi \int_a^b [f(x)]^2\,dx

Disk method (rotation about the x-axis)

V = \pi \int_a^b \big([f(x)]^2 - [g(x)]^2\big)\,dx

Washer method (region between two curves f ≥ g)

V = 2\pi \int_a^b x\,f(x)\,dx

Shell method (rotation about the y-axis, region under f(x) ≥ 0 on [a,b] ≥ 0)

Applications

Computing the volume and mass of manufactured parts with rotational symmetry (e.g. lathe-turned components, pressure vessels) uses volumes of revolution.

Worked Examples

Use the disk method: V = π∫[f(x)]² dx with f(x)=√x.
$V = \pi\int_0^4 (\sqrt{x})^2\,dx = \pi\int_0^4 x\,dx$
Evaluate the integral.
$= \pi\left[\frac{x^2}{2}\right]_0^4 = \pi\cdot 8$

Answer: V = 8π

Practice Problems

Difficulty 5/10

Find the volume generated by rotating y=x² from x=0 to x=2 about the x-axis.

Difficulty 6/10

Use the shell method to find the volume from rotating the region bounded by y=x², y=0, x=1 about the y-axis.

Difficulty 6/10

A storage tank is formed by rotating the region under y = 2 (a horizontal line) from x = 0 to x = 5 about the x-axis, giving a cylinder. Use the disk method to find its volume (units in metres).

Common Mistakes

Common Mistake

Using the disk method (single radius) when the region doesn't touch the axis of rotation, ignoring the inner radius.

If there's a gap between the region and the axis, use the washer method with both an outer and inner radius, not just the disk method.

Common Mistake

Mixing up which variable to integrate with respect to when switching between disk/washer (integrate along the axis of rotation) and shell method (integrate perpendicular to the axis).

For rotation about the x-axis, disk/washer integrates dx; for rotation about the y-axis using shells, integrate dx as well but with x as the shell radius — always carefully identify which variable represents distance from the axis versus position along it.

Quiz

The DISK method computes a volume of revolution as:

π∫[R(x)]² dx2π∫x·f(x) dx∫f(x) dxπ∫R(x) dx

The SHELL method (V = 2π∫x·f(x) dx) is often preferred when:

rotating about an axis parallel to the strips, so cylindrical shells avoid awkward disk radiithe region is a rectanglethere is no axis of rotationthe function is constant

Summary

Disk method: V = π∫[f(x)]²dx, for a region touching the axis of rotation.
Washer method: V = π∫([f(x)]²-[g(x)]²)dx, for a region with a gap between it and the axis.
Shell method: V = 2π∫x f(x)dx, useful when rotating about an axis parallel to the direction of integration where disks would require solving for x in terms of y.
Choose the method that avoids having to invert the function algebraically.

References

WebsiteWikipedia — Solid of revolution