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.
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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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.
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.
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.
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.
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).
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).
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.
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.