Volume of Revolution Calculator (Disk & Shell)

Turning a curve into a solid

Spin a flat region around a straight line and it sweeps out a three-dimensional solid. A rectangle becomes a cylinder, a right triangle becomes a cone, a semicircle becomes a sphere. Calculus lets you find the volume of any such solid, even when the generating curve is messier than a straight line. There are three standard techniques, and this calculator handles the two most common: the disk method and the shell method.

Disk method (revolving around the x-axis)

V = π ∫ from a to b of [f(x)]² dx

Picture slicing the solid into thin coins stacked along the x-axis. Each coin is a disk of radius f(x) and thickness dx, so its volume is π[f(x)]² dx. The integral adds them all up. Use this when the region sits right against the axis with no hole through the middle.

Shell method (revolving around the y-axis)

V = 2π ∫ from a to b of x · f(x) dx

This time imagine the solid as a set of nested cylindrical shells, like the rings of a tree. A shell at position x has radius x, height f(x), and thickness dx, giving 2πx·f(x) dx. The shell method is the easy choice when revolving around a vertical axis, because you avoid having to rewrite the curve as x in terms of y. Shell integrals usually want a lower bound of zero or more, since the radius x should not go negative.

The washer method

When the region lies between two curves, revolving it leaves a hole and each slice is a washer instead of a solid disk: V = π ∫ ([R(x)]² − [r(x)]²) dx, with R the outer radius and r the inner. You can get that result here by running the disk method twice, once for the outer curve and once for the inner, then subtracting.

A quick check

Revolve y = √x from x = 0 to x = 4 around the x-axis. The disk method gives V = π ∫ x dx from 0 to 4 = 8π, about 25.13. The same value falls out of the shell method applied to y = x² from 0 to 2, a tidy coincidence worth verifying yourself.

Where it matters

Any rotationally symmetric object has a volume you can find this way: turned table legs, wine barrels, rocket nose cones, ball bearings, and the goblet from every calculus final.