Volume of Revolution Calculator

Calculate the volume of a solid of revolution by entering your function(s), lower limit, upper limit, and axis of rotation. Choose between the disk/washer (rings) method or the shell (cylinders) method — the calculator evaluates the definite integral numerically and returns the total volume in cubic units. Also try the calculate Average Rate of Change (Δy/Δx), Change in y (Δy = y₂ − y₁) & Change in x (Δx = x₂ − x₁) — Average Rate of Change.

Enter the outer boundary curve as a function of x. Use ^ for powers, sqrt() for square root.

Enter the inner boundary curve. Use 0 if rotating a single curve around the axis.

Lower bound of integration.

Upper bound of integration.

Enter the value of y for the horizontal axis of rotation (e.g. 0 for x-axis, 1 for y=1).

Disk/Washer integrates π∫[R²−r²]dx. Shells integrates 2π∫x·f(x)dx.

Results

Total Volume

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Volume (× π)

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Integral Value (without π)

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Method Used

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Results Table

Frequently Asked Questions

What is a solid of revolution?

A solid of revolution is a 3D shape formed by rotating a 2D region or curve around a straight axis (usually the x-axis or y-axis). Common examples include spheres, cones, cylinders, and toruses, all of which can be generated by rotating a simple curve. See also our use the Simpson's Rule Calculator.

What is the disk/washer (rings) method?

The disk/washer method computes volume by integrating cross-sectional areas perpendicular to the axis of rotation. For a single curve f(x) rotated about y = c, the formula is V = π∫[a to b] (f(x) − c)² dx. When there's an inner boundary g(x), the washer formula subtracts the inner radius: V = π∫[a to b] [(f(x)−c)² − (g(x)−c)²] dx.

What is the shell/cylinder method?

The shell method integrates cylindrical shells parallel to the axis of rotation. For rotation around y = c (horizontal axis), the formula is V = 2π∫[a to b] |x − c| · f(x) dx. It is often easier to use than the disk method when f(x) is difficult to invert.

When should I use the disk method vs the shell method?

Use the disk/washer method when the cross-sections perpendicular to the axis are easy to express. Use the shell method when the function is easier to integrate in its current form without inverting. Both methods yield the same result for the same region. You might also find our Double Integral Calculator useful.

How do I rotate around an axis other than the x-axis?

Enter the y-value of your axis in the 'Axis of Rotation' field. For example, to rotate around y = 1, enter 1. The calculator adjusts the radius calculations accordingly — outer radius becomes (f(x) − axis) and inner radius becomes (g(x) − axis).

What functions can I enter?

You can use standard mathematical notation: x^2 for x squared, sqrt(x) for square root, sin(x), cos(x), tan(x), exp(x), log(x), abs(x), and constants like pi and e. Use * for multiplication where needed (e.g. 2*x).

What if my two curves intersect?

If your curves intersect within [a, b], the 'outer' and 'inner' roles may swap mid-interval, which can affect the result. Make sure f1(x) ≥ f2(x) throughout the interval [a, b] for the disk/washer method to give a correct answer. Consider splitting the interval at the intersection point.

Why does this calculator use numerical integration?

Many functions do not have closed-form antiderivatives, so this calculator uses Simpson's Rule with a high number of subdivisions (n = 10000) to evaluate the definite integral numerically. For most practical functions, this gives results accurate to 5–6 decimal places.