V↻

Volume of Revolution Calculator with Graph

Calculus & GraphingFreeStep by stepGraph

Rotating a curve around the x-axis sweeps out a solid. Its volume is found with the disc method: each thin slice is a disc of radius f(x), so its area is πf(x)², and integrating adds the slices up. This calculator gives the volume exactly in terms of π when it can, checks it numerically and draws the solid's outline.

Try:

How to use the volume of revolution calculator

  1. Type the function y = f(x).
  2. Enter the start and end of the interval.
  3. Press Calculate to see the volume and the outline.

Formula

V=π∫abf(x)2 dxV = \pi \int_a^b f(x)^2\,dx

Worked example: y = √x, 0 to 4

  • y = f(x): sqrt(x)
  • From x =: 0
  • To x =: 4

✓ Answer checked

V=8π≈25.132741V = 8\pi \approx 25.132741
−112345−2−112y = f(x)
Spinning the curve around the x-axis sweeps out a solid; this is its side-on outline. Each thin slice is a disc of area π·f(x)².
Step-by-step working (3 steps)

  1. Disc method Disc method

    Each slice is a disc of radius f(x), so its area is π f(x)². Add the slices with an integral.

    V=π∫04(x)2dxV = \pi\int_{0}^{4} \left(\sqrt{x}\right)^2 dx

  2. Square and integrate

    Integrate f(x)².

    ∫x dx=8\int x\,dx = 8

  3. Multiply by π

    Keep π for the exact answer.

    V=8π≈25.132741V = 8\pi \approx 25.132741
Formulas used
V=π∫abf(x)2 dxV = \pi\int_a^b f(x)^2\,dx
How this was checked
  • ✓ Numerical integration of π f(x)² gives the same volume.

Rotation about the x-axis.

Frequently asked questions

What is the volume when y = √x from 0 to 4 is rotated?

π ∫ x dx from 0 to 4 = 8π ≈ 25.13.

How does this give the volume of a cone?

Rotating y = x/2 from 0 to 6 makes a cone of radius 3 and height 6, with volume 18π, matching ⅓πr²h.

Which axis is used?

The x-axis. For rotation about the y-axis, rewrite the curve as x in terms of y.

People also search for

volume of revolution calculatordisc method calculatorsolid of revolutionvolume of revolution calculator with graphvolume of revolution calculator with stepsfree volume of revolution calculator online

Related calculators

∫ Integral CalculatorFind indefinite and definite integrals with steps: power rule, substitution and integration by parts. Graph V Volume CalculatorFind the volume of a cube, cuboid, cylinder, cone, sphere, hemisphere or square pyramid. ∫f−g Area Between Curves CalculatorFind the area between two curves over an interval. Graph

More calculus & graphing tools

d/dx Derivative CalculatorDifferentiate any function with every rule shown: power, product, quotient and chain rules. Graph lim Limit CalculatorEvaluate limits step by step: direct substitution, L'Hôpital's rule for 0/0, and limits at infinity. Graph f(x) Graphing CalculatorPlot up to three functions with an animated graph. Graph Σ Taylor Series CalculatorFind the Taylor or Maclaurin polynomial of a function up to order 8, with exact coefficients and a graph of the approximation hugging the curve. Graph