Volume of Revolution Calculator
Volumes of solids of revolution by the disc, washer and shell methods, and surface areas of revolution, with answers recognised as exact multiples of π where they are.
About the Volume of Revolution Calculator
Rotating y = f(x), a ≤ x ≤ b, about the x-axis sweeps out a solid of volume π∫y² dx (the disc method); with a hole of radius g(x) it is π∫(f² − g²) dx (washers). About the y-axis, the shell method gives 2π∫x·f(x) dx.
The curved surface area about the x-axis is 2π∫|y|√(1 + y′²) dx.
The formula
V = π∫ₐᵇ y² dx; S = 2π∫ₐᵇ y√(1 + y′²) dx
Worked example
y = √x from 0 to 4, about the x-axis
- V = π∫₀⁴ x dx = π[x²/2]₀⁴
- = 8π ≈ 25.13
Questions people ask
Disc or shell?
Either gives the same volume. Discs slice across the axis of rotation; shells slice parallel to it, which is often easier about the y-axis.
Why isn't my answer a multiple of π?
It is always π times an integral, but that integral may not be a simple fraction; then the decimal is shown.