Shell Method Volume Calculator

Work out shell method volume instantly with clear inputs, formula shown and shareable results.

About Shell Method Volume Calculator

The shell method slices the solid into thin cylindrical shells. Each has circumference 2πx, height f(x) and thickness dx, so the volume integrates 2πx·f(x).

Formula

Shell method

V = 2π ∫ₐᵇ x · f(x) dx

Worked example

With the default values below, Shell Method Volume Calculator returns Volume by shells of 23.56194490. Change any input above and every figure here is recalculated in your browser.

Inputs used

Height function coefficients: 1, 0, 0
From x =: 1
To x =: 2

Results

Volume by shells: 23.56194490
Disc method about the x-axis for comparison: 19.47787445
Radius of the outermost shell: 2.000000
Height of the outermost shell: 4.000000

Step by step

1Values used
2Shell method
3Volume by shells= 23.56194490
4Disc method about the x-axis for comparison= 19.47787445
5Radius of the outermost shell= 2.000000
6Height of the outermost shell= 4.000000

Assumptions and limits

The result is derived only from the 3 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 = x² from 0 to 2 about the y-axis.

2π∫x³ dx = 2π × 4 = 8π ≈ 25.1327412.

Why is the radius x and not y?

Because the shells are centred on the y-axis, so their radius is the horizontal distance from that axis.