Volume of Revolution Calculator (Disk, Washer and Shell Methods)
Find the volume of a solid of revolution: rotate y = f(x), or the region between f(x) and g(x), about the x-axis (disk/washer) or the y-axis (shell method) over [a, b].
How it works
Disk/washer (about the x-axis): V = π ∫ₐᵇ [f(x)² − g(x)²] dx · Shell (about the y-axis): V = 2π ∫ₐᵇ x·[f(x) − g(x)] dx
Rotating a region around an axis sweeps out a solid. Slicing perpendicular to the x-axis gives disks of radius f(x) — or washers with a hole of radius g(x) — so the volume is π times the integral of the outer radius squared minus the inner radius squared. Rotating about the y-axis is easiest with cylindrical shells of radius x and height f(x) − g(x). The integral is evaluated with Simpson’s rule on 2,000 subintervals.
Worked example
Rotate y = x² from x = 0 to 2 about the x-axis: V = π ∫₀² (x²)² dx = π · x⁵/5 |₀² = 32π/5 = 6.4π ≈ 20.106193. About the y-axis the shell method gives 2π ∫₀² x · x² dx = 8π ≈ 25.132741.
Sources: OpenStax Calculus Volume 1 — 6.2 Determining Volumes by Slicing · OpenStax Calculus Volume 1 — 6.3 Volumes of Revolution: Cylindrical Shells
Frequently asked questions
When do I use the washer method instead of the disk method?
When the region does not touch the axis of rotation, so the solid has a hole. The washer area is π(R² − r²) with outer radius R = f(x) and inner radius r = g(x).
When is the shell method easier?
When you rotate around the y-axis but the curve is given as y = f(x). Shells avoid solving for x in terms of y.
Why is there a π in every answer?
Each slice is a circle (area πr²) or a cylindrical shell (circumference 2πx), so π factors out of the integral. The calculator also shows the volume as a multiple of π.
Can I rotate around a line like y = 2?
Shift the functions first: about y = 2 use radii f(x) − 2 and g(x) − 2, i.e. enter f(x) − 2 and g(x) − 2 as the curves.
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This calculator is for educational purposes. Double-check important results before acting on them.