Volume Of Revolution Calculator

Work out volume of revolution instantly with clear inputs, formula shown and shareable results.

About Volume Of Revolution Calculator

Revolving about the x-axis stacks discs of radius f(x), giving V = π∫f(x)² dx. Revolving about the y-axis sweeps out cylindrical shells of radius x and height f(x), giving V = 2π∫x·f(x) dx.

Formulas

Disc method

V = π ∫ₐᵇ [f(x)]² dx

Shell method

V = 2π ∫ₐᵇ x·f(x) dx

Worked example

With the default values below, Volume Of Revolution Calculator returns Volume as configured of 7.33038286. Change any input above and every figure here is recalculated in your browser.

Inputs used

Curve coefficients, highest power first: 1, 0
From x =: 1
To x =: 2
Revolve aboutone of the x-axis (disc method), the y-axis (shell method): the x-axis (disc method)

Results

Volume as configured: 7.33038286
Disc method about the x-axis: 7.33038286
Shell method about the y-axis: 14.66076572
Surface area of revolution: 13.32864881

Step by step

1Values used
2Disc method
3Shell method
4Volume as configured= 7.33038286
5Disc method about the x-axis= 7.33038286
6Shell method about the y-axis= 14.66076572
7Surface area of revolution= 13.32864881

Assumptions and limits

The result is derived only from the 4 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

Revolve y = x from 0 to 2 about the x-axis.

π∫x² dx = 8π/3 ≈ 8.3775804, the cone of radius 2 and height 2.

Which method should I choose?

Whichever avoids solving for the other variable. Discs suit rotation about the axis of the variable you integrate over.