Math · Calculus II · Concept

Volumes of revolution: disks, washers and shells

Turning a region about a line sweeps out a solid of revolution. Slice it perpendicular to the axis and each slice is a disk or a washer, with area πR² or π(R² − r²); slice it parallel to the axis and each piece is a cylindrical shell. Adding up the slices with an integral gives the volume.

Volume by slicing

If every cross section perpendicular to the x-axis has area A(x), a thin slice of thickness dx has volume about A(x) dx, and the integral adds the slices. Disks, washers and shells are ways of finding A(x) or its equivalent for a solid of revolution.

V = \int_{a}^{b} A (x) d x

The disk method

When the region under y = f(x) turns about the x-axis, each perpendicular slice is a disk whose radius is the height of the curve.

V = \int_{a}^{b} π {[f (x)]}^{2} d x

The washer method

When the region lies between two curves, each slice is a washer: a disk of outer radius R(x) with a hole of inner radius r(x). Subtract the two areas, not the two radii.

V = \int_{a}^{b} π (R {(x)}^{2} - r {(x)}^{2}) d x

The shell method

Turning the region about the y-axis, a thin vertical strip at x sweeps out a cylindrical shell of radius x, height h(x) and thickness dx. Cut and unrolled, the shell is nearly a flat slab of volume 2πx·h(x)·dx.

V = \int_{a}^{b} 2 π x h (x) d x

Choosing a method

Disks and washers slice perpendicular to the axis, so they integrate along the axis; shells slice parallel to it. Choose the one whose radius and height are easier to write. For a region given by y = f(x) turned about the y-axis, shells avoid solving for x.

Common mistakes

Using π(R − r)² for a washer instead of π(R² − r²).
Forgetting to square the radius in the disk method.
Integrating in x when the slices are perpendicular to the y-axis.
Measuring the radius from the origin instead of from the axis of rotation when the axis is not a coordinate axis.

Key terms

Solid of revolution: The solid swept out when a plane region turns about a line in its plane. Its volume can be found by slicing it into disks, washers or cylindrical shells.
Disk method: Finding the volume of a solid of revolution by slicing perpendicular to the axis: each slice is a disk of radius R(x), so V = ∫πR(x)² dx.
Washer method: The disk method for a region that does not touch the axis: each slice is a washer with outer radius R and inner radius r, so V = ∫π(R² − r²) dx.
Shell method: Finding the volume of a solid of revolution by slicing parallel to the axis: a strip at distance r from the axis with height h sweeps out a cylindrical shell of volume about 2πrh times its thickness.
Definite integral: ∫ₐᵇ f(x) dx: the signed area between the graph of f and the x-axis from a to b, defined as the limit of Riemann sums. Area below the axis counts as negative.

Work through an example

The region under y = √x from x = 0 to x = 4 is revolved about the x-axis. Find the volume of the solid.

Find a volume with the disk method →

Find a volume with the washer method →

Find a volume with the shell method →

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