Volume of Revolution Calculator
This calculator finds the volume of the solid generated when the region under a curve y = f(x) is revolved about the x-axis, using the disk method.
Function f(x)
Solid generated by revolving the curve about the x-axis
How does this calculator work?
The volume of a solid of revolution about the x-axis is V = π·∫_a^b f(x)² dx (disk method). Choose a function and the limits a and b; this tool evaluates the integral numerically with Simpson’s rule and returns the volume in cubic units, plus a plot of the curve being revolved.
Formula
How this is calculated
Pick a function f(x) and the interval limits a and b. The region bounded by the curve, the x-axis, and the vertical lines x = a and x = b is revolved a full turn about the x-axis. Each thin slice perpendicular to the axis becomes a disk of radius f(x) and thickness dx, so its volume is π·f(x)²·dx. Summing these gives V = π·∫_a^b f(x)² dx.
The definite integral of the squared function is evaluated numerically with composite Simpson’s rule using n = 1000 subintervals, which gives a highly accurate result for smooth functions. The calculator handles a > b by integrating over the ordered interval, and the reported volume is always non-negative because the integrand f(x)² is non-negative.
Volumes are expressed in cubic units consistent with the units of x and f(x). The disk method assumes the region touches the axis (radius = f(x)); for regions offset from the axis use the washer method instead. Functions with singularities inside the interval (for example 1/x near x = 0) can produce non-finite or unreliable values, in which case no result is shown.
Frequently asked questions
It slices the solid into thin circular disks perpendicular to the axis of revolution. Each disk has radius f(x) and thickness dx, giving volume π·f(x)²·dx; integrating from a to b sums them.
The integral is computed with composite Simpson’s rule on 1000 subintervals, which is very accurate for smooth curves. Functions with sharp features or singularities in the interval reduce accuracy.
The integrand is f(x)², which is never negative, so the integral and the resulting volume are always greater than or equal to zero regardless of the sign of f(x).
Also known as
TG we-Calculate Editorial Team. (2026). Volume of Revolution Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/volume-of-revolution-calculator
TG we-Calculate Editorial Team. "Volume of Revolution Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/volume-of-revolution-calculator.
TG we-Calculate Editorial Team, "Volume of Revolution Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/volume-of-revolution-calculator
@misc{wecalculate_volume_of_revolution_calculator, title = {Volume of Revolution Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/volume-of-revolution-calculator}}, year = {2026}, note = {TG we-Calculate} }
Did this calculator help you?