Volume of Revolution Calculator

The volume of revolution calculator computes the volume of a 3D solid formed by rotating a curve around an axis, using the disc, washer, or shell method. It is used by calculus students, engineers designing rotational components, and anyone working with volumes of symmetric solids. Enter your function, the axis of rotation, and the interval to get the volume with full working.

How to Use the Volume of Revolution Calculator

1. Enter your function f(x) in the Volume of Revolution Calculator input field. Choose the axis of rotation: x-axis, y-axis, or a horizontal or vertical line such as y = 2.
2. Select the method: disc method for solid regions, washer method for hollow regions, or shell method.
3. Set the lower bound a and upper bound b of the interval.
4. Click Calculate to evaluate the integral and display the volume.

The Formula

Disc method (rotating y = f(x) around the x-axis):

V = pi * Integral from a to b of [f(x)]^2 dx

Washer method (rotating the region between y = f(x) and y = g(x) around the x-axis, with f above g):

V = pi * Integral from a to b of ([f(x)]^2 - [g(x)]^2) dx

Shell method (rotating y = f(x) around the y-axis):

V = 2*pi * Integral from a to b of x * f(x) dx

Each method slices the solid differently. Discs and washers slice perpendicular to the axis; shells slice parallel to the axis. Choose the method that results in the simpler integral.

Real-World Example

Find the volume of the solid formed by rotating f(x) = sqrt(x) around the x-axis on [0, 4] using the disc method.

1. Set up the integral: V = pi * Integral from 0 to 4 of (sqrt(x))^2 dx = pi * Integral from 0 to 4 of x dx
2. Integrate: V = pi * [x^2/2] from 0 to 4
3. Evaluate: V = pi * (8 - 0) = 8*pi

The volume is 8 pi cubic units, approximately 25.13 cubic units. This solid looks like a paraboloid of revolution, narrow at the origin and widening toward x = 4.

Choosing Between Disc, Washer, and Shell Methods

Use the disc method when the region being rotated touches the axis of rotation with no hole in the middle. Use the washer method when there is a gap between the inner function and the axis, creating a hollow solid like a donut slice. Use the shell method when rotating around the y-axis and the function is expressed in terms of x, because it avoids having to invert the function to express x in terms of y. In practice, both the washer and shell methods can solve the same problem; choose whichever leads to the easier integral.

Frequently Asked Questions

What is the difference between the disc and washer methods? The disc method applies when the region being rotated is bounded directly by the axis of rotation, producing a solid cross-section. The washer method applies when there is an inner boundary function, creating an annular (ring-shaped) cross-section with a hole in the centre.

Can I rotate around lines other than the x or y axis? Yes. For rotation around y = k, replace f(x) in the disc formula with [f(x) - k]. For rotation around x = k, use the shell method with (x - k) in place of x. The principle is the same: measure the radius of each disc or shell from the axis.

Why does pi appear in the volume formula? Each cross-sectional slice is a circular disc with area pi * r^2. Integrating these disc areas along the axis gives the total volume, with pi as a constant factor throughout.

How does volume of revolution relate to surface area of revolution? They are related but distinct. The volume integral sums the areas of cross-sectional discs, while the surface area integral sums the circumferences of rings formed by the curve as it rotates. The surface area formula uses the arc length element rather than the squared radius.

One solid measured by two methods

Take the region under f(x) = sqrt(x) on the interval from 0 to 4 and rotate it about the y-axis.

By shells, which is the tool's own route: V = 2 pi times the integral from 0 to 4 of x times sqrt(x) dx. The integrand is x to the power 3/2, whose antiderivative is two fifths of x to the power 5/2. At x = 4 that is two fifths of 32, which is 12.8. So V = 2 pi times 12.8, which is 25.6 pi, and the tool prints 80.42477.

By washers, worked by hand because the tool has no washer setting: slice perpendicular to the y-axis instead. The outer radius of each slice is the line x = 4 and the inner radius is x = y squared, so the slice area is pi times (16 minus y to the fourth). Integrate that from y = 0 to y = 2, which is the range y = sqrt(x) covers as x runs from 0 to 4. The result is pi times (32 minus 32 fifths), which is pi times 25.6, or 80.4247719.

The two routes describe the same solid and land on the same figure to eight significant figures. That agreement is the most useful check on any volume of revolution: when two set-ups differ by more than the numerical error, one of them is wrong.

What the numerical integration returns on the built-in examples

The tool evaluates every integral by Simpson's rule with 2,000 subintervals. The table below checks each of the tool's own example buttons against its closed form.

The last row carries the widest gap, 1.6 times 10 to the minus 8, and the square root is the reason rather than the interval. The integrand x to the power 3/2 has a second derivative that grows without bound as x approaches zero, and no fixed-step rule handles that edge as cleanly as it handles a polynomial over the same range. The printed figure is still right to the eight digits the tool shows.

Which cases each setting covers

The disc button rotates about the x-axis and squares f(x) slice by slice. The shell button rotates about the y-axis and multiplies each shell's circumference by its height. Between them they cover a solid region with no hole and a rotation about one of the two axes.

Three cases in the steps above have nowhere to go in the tool. There is no axis field, so rotation about y = 2 or x = 3 cannot be entered. There is no washer setting, so a region with an inner boundary cannot be set up at all. And the introduction above promises the volume with full working, while the result panel prints the volume, the method, the formula and the interval with no intermediate line. The numbered steps also run 1, 3, 4, 5, with no step 2 anywhere on the page.

The bounds are checked. A lower bound that is not smaller than the upper bound stops the run and prints "Enter valid bounds with a < b." An expression the parser cannot read comes back as a message asking for it to be checked.

What the integral assumes

Disc integration squares the radius, so a function that dips below the axis returns the same volume as its reflection above it. The tool cannot tell the two apart, and a shape that actually crosses the axis has a different volume from either. Watch for that on a function such as sin(x) over an interval longer than pi, where part of the region lies on the other side of the axis.

The expression box takes a limited vocabulary: powers written with the caret, square root, the trig functions, abs, ln, pi and e, and a product such as 2x written without a sign between the number and the letter. Anything outside that set evaluates to a value that is not a number, and the run stops with an error rather than a wrong volume.

Where the disc and shell relations come from

The shell formula comes from the surface area of a cylinder, 2 pi r h, summed over the shells between the two bounds. The disc formula comes from the area of a circle, pi r squared, summed over the slices along the axis. Both are standard in an introductory calculus course and both follow from the definite integral as a limit of Riemann sums. The numerical route in the tool is the composite Simpson's rule, which is exact for any polynomial up to degree three and very close for smooth functions above that.

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