Volume of Revolution Calculator

Calculate the volume of a solid of revolution using the disk, washer, and shell methods. Enter your function, bounds, and axis of rotation to get step-by-step solutions with interactive 3D visualizations.

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About Volume of Revolution Calculator

The Volume of Revolution Calculator computes the volume of a three-dimensional solid formed by rotating a two-dimensional region around an axis. This is one of the most important applications of integral calculus, widely used in engineering, physics, and manufacturing to determine volumes of objects with rotational symmetry — from engine cylinders to satellite dishes.

Three Methods of Calculation

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Disk Method

Stack infinitely thin circular disks perpendicular to the axis. Best for single curves rotated around the x-axis: \( V = \pi \int_{a}^{b} [f(x)]^2 \, dx \)

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Washer Method

Like the disk method but with a hole. For regions between two curves: \( V = \pi \int_{a}^{b} [R(x)^2 - r(x)^2] \, dx \)

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Shell Method

Wrap cylindrical shells around the axis. Ideal for rotation around the y-axis: \( V = 2\pi \int_{a}^{b} x \cdot f(x) \, dx \)

How to Use the Volume of Revolution Calculator

Choose a method — Select Disk, Washer, or Shell depending on your problem setup.
Enter function(s) — Type your function f(x) using standard math notation (e.g., x^2, sqrt(x), sin(x)). For the Washer method, also enter the inner function g(x).
Set bounds — Enter the lower bound (a) and upper bound (b) of integration.
Select the axis of rotation — Choose from x-axis, y-axis, or enter a custom axis value.
Click Calculate Volume — View the result with step-by-step MathJax formulas, an interactive 3D visualization, and a comparison across all three methods.

When to Use Each Method

Scenario	Best Method	Why
Single curve rotated around x-axis	Disk	Simplest setup — only need f(x)
Region between two curves, rotated around x-axis	Washer	Naturally handles outer and inner radii
Curve rotated around y-axis	Shell	Avoids inverting f(x) to express x as a function of y
Function hard to invert	Shell	No need to solve for x in terms of y
Axis of rotation is horizontal	Disk/Washer	Disks are perpendicular to horizontal axis
Axis of rotation is vertical	Shell	Shells wrap around vertical axis naturally

Common Examples

Shape	Function	Bounds	Volume
Cone	f(x) = x	[0, r]	\( \frac{1}{3}\pi r^3 \)
Sphere	f(x) = √(r² − x²)	[−r, r]	\( \frac{4}{3}\pi r^3 \)
Paraboloid	f(x) = √x	[0, h]	\( \frac{1}{2}\pi h^2 \)
Torus (ring)	Washer with offset axis	Circle	\( 2\pi^2 R r^2 \)

Supported Functions

This calculator accepts a wide range of mathematical expressions:

Polynomials: x^2, x^3 + 2x, 3x^2 - x + 1
Trigonometric: sin(x), cos(x), tan(x)
Inverse trig: asin(x), acos(x), atan(x)
Exponential/Log: exp(x), ln(x), log(x)
Root: sqrt(x)
Constants: pi, e
Combinations: x^2 * sin(x), sqrt(x) + 1

Frequently Asked Questions

What is the volume of revolution?

The volume of revolution (or solid of revolution) is the volume of a 3D object created by rotating a 2D curve or region around an axis. It is calculated using integral calculus with the disk, washer, or shell method. Common real-world examples include bottles, bowls, vases, and engine pistons.

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

Use the disk method when the axis of rotation is perpendicular to the variable of integration (typically rotating around the x-axis with functions of x). Use the shell method when the axis of rotation is parallel to the variable of integration (typically rotating around the y-axis with functions of x). The shell method is often easier when the function is hard to invert.

What is the washer method?

The washer method is an extension of the disk method for regions bounded by two curves. It subtracts the volume of the inner solid from the outer solid using the formula \( V = \pi \int_{a}^{b} [R(x)^2 - r(x)^2] \, dx \), where R(x) is the outer radius and r(x) is the inner radius.

How do I choose the axis of rotation?

The most common axes are the x-axis (y = 0) and y-axis (x = 0). You can also rotate around any horizontal line y = k or vertical line x = k. The choice of axis affects which method is most convenient and changes the radius expressions in the integral.

What functions does this calculator support?

This calculator supports polynomials (x^2, x^3), trigonometric functions (sin, cos, tan), exponential and logarithmic functions (exp, ln, log), square root (sqrt), and combinations with standard arithmetic operators. Use x as the variable.

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"Volume of Revolution Calculator" at https://MiniWebtool.com/volume-of-revolution-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: 2026-04-04

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Volume of Revolution Calculator

About Volume of Revolution Calculator

Three Methods of Calculation

How to Use the Volume of Revolution Calculator

When to Use Each Method

Common Examples

Supported Functions

Frequently Asked Questions

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