Hard

Volume of Revolution: The Washer Method

Simple Explanation

When the region being rotated is trapped between two curves (with a gap between the region and the axis of rotation), each cross-section is a "washer" — a disk with a smaller disk removed from its center. The volume is V = π∫ₐᵇ([R(x)]²−[r(x)]²)dx, where R is the outer radius and r is the inner radius.

Why Do We Need It?

Many real solids of revolution have a hole through them — pipes, rings, washers, tubes — and the washer method is exactly the tool for computing their volume.

See It

The region between y = x and y = x², from x = 0 to x = 1, before rotation

Formula

Volume of Revolution: The Washer Method

V = π ∫ₐᵇ ( [R(x)]² − [r(x)]² ) dx

Rotating a region between two curves around an axis sweeps out a solid with a hole through it; each cross-section is a "washer" — a disk of outer radius R(x) with a smaller disk of inner radius r(x) removed.

R(x): — the outer radius — distance from the axis to the farther boundary curve
r(x): — the inner radius — distance from the axis to the nearer boundary curve

When to use it: Whenever the region between two curves (with a gap between the region and the axis of rotation) is revolved around that axis.

Worked Example

Find a volume of revolution using the washer method

Find the volume generated by rotating the region between y=x (outer) and y=x² (inner), from x=0 to x=1, around the x-axis.

Why Does This Work?

Each washer's cross-sectional area is the area of the outer disk minus the area of the inner disk removed from its center: π[R(x)]² − π[r(x)]² = π([R(x)]²−[r(x)]²) — summing these thin washer volumes across [a,b] via integration gives the total volume of the hollow solid.

Real-Life Example

Computing the volume of a metal washer or pipe segment

A machinist manufactures a ring-shaped washer or a pipe segment, whose cross-section is an annulus (a disk with a smaller disk removed) rather than a solid disk.

The washer method computes exactly how much material such a hollow, ring-shaped part requires, directly from its inner and outer radius functions.

Practice

Find the volume when the region between R(x)=2 (outer) and r(x)=x (inner), from x=0 to x=2, is rotated around the x-axis.

Hard

Common mistake

Squaring the difference of the radii, (R(x)−r(x))², instead of subtracting the squares, [R(x)]²−[r(x)]² — these are not the same, and only the second one correctly represents "outer disk area minus inner disk area."

Quick Review

V = π∫ₐᵇ ([R(x)]²−[r(x)]²) dx — outer radius squared minus inner radius squared.
Used when the region being rotated does not touch the axis of rotation directly.
Square each radius separately — never square their difference.