Volume of Solids of Revolution: Disk & Shell Method

30 September 2026Mathematics Try the interactive 3D model →

A solid of revolution is the three-dimensional shape swept out when a flat region is spun around a straight line. Spinning the area under y = √x around the x-axis carves out a shape like a pointed bowl, and calculus finds its exact volume.

This matters because countless real objects are rotationally symmetric: vases, bottles, domes, wheels, and machine parts turned on a lathe. Two standard methods, disks (or washers) and cylindrical shells, reduce any such volume to a single integral.

Key Concepts

Axis of rotation: the line the region spins around. Slices are cut perpendicular to this axis for the disk and washer methods, and parallel to it for the shell method.
Disk method: each thin slice perpendicular to the axis is a solid circular disk of radius R(x) and thickness dx, with volume π(R(x))² dx.
Washer method: when the region does not touch the axis, each slice is a disk with a hole: outer radius R(x), inner radius r(x), volume π(R(x)² − r(x)²) dx.
Shell method: each slice parallel to the axis becomes a thin cylindrical shell of radius x, height h(x), and circumference 2πx, with volume 2πx · h(x) dx.
Choosing a method: pick whichever gives the simpler integral. Spinning around the x-axis usually favours disks with dx; spinning around the y-axis often favours shells with dx.

Formulas

Disk about the x-axis: V = π∫(R(x))² dx from x = a to x = b, in cubic units.
Washer about the x-axis: V = π∫(R(x)² − r(x)²) dx from x = a to x = b.
Shell about the y-axis: V = 2π∫x · h(x) dx from x = a to x = b, where h(x) is the height of the region at position x.
Disk about the y-axis: V = π∫(R(y))² dy from y = c to y = d, slicing horizontally instead.

Worked Example

Find the volume when the region under y = √x, from x = 0 to x = 4, is revolved about the x-axis.

Choose the method: slices perpendicular to the x-axis are disks touching the axis, so use the disk method with dx.
Identify the radius: each disk has radius R(x) = √x, the height of the curve above the axis.
Set up the integral: V = π∫₀⁴ (√x)² dx = π∫₀⁴ x dx.
Integrate: ∫ x dx = x²/2. Evaluating from 0 to 4: π(16/2 − 0) = π × 8 = 8π ≈ 25.13 cubic units.
Sanity check: the enclosing cylinder of radius 2 and length 4 has volume π × 2² × 4 = 16π ≈ 50.27. Our bowl-like solid is slimmer than the cylinder, and 8π is exactly half of 16π, which is plausible.

Common Misconceptions

Squaring after integrating. The radius must be squared inside the integral: π ∫ (R(x))² dx, never π(∫ R(x) dx)².
Mixing up the 2π factor. The 2π belongs only to the shell method (circumference of the shell); the disk method uses just π (area of the circular face).
Using the function as radius when the axis is shifted. Rotating around y = −1 makes the radius f(x) + 1, not f(x). Radius is always the distance from the curve to the axis.

Real-World Applications

Manufacturing: volumes of parts shaped on a lathe, such as axles, bowls, and decorative spindles.
Containers: capacity of rotationally symmetric vessels like vases, bottles, and storage tanks designed from a profile curve.
Architecture: volumes of domes and curved towers generated by revolving an arch profile.
Medicine: estimating organ volumes from cross-sectional scan profiles in imaging analysis.

Volumes of revolution come from stacking simple shapes: disks or washers cut across the axis, or shells wrapped around it. Square the radius before integrating, keep the axis-to-curve distance honest, and one integral delivers the exact volume of any spun shape.