Volume of Revolution Calculator
MiniWebtool’s Volume of Revolution Calculator models solids formed by rotating curves. Choose disk, washer, or shell method, supply functions and bounds, and inspect the integral setup with a three-dimensional visualization.
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Choose the disk, washer or shell method below and enter the function, the bounds and the axis to get the volume with step-by-step working and a 3D view of the solid.
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.
At a Glance
Volume of Revolution Calculator finds the volume of a solid formed by rotating a region around an axis, using the disk, washer or shell method. You enter functions and bounds and see the integral setup.
| Feature | What you get | Good for |
|---|---|---|
| Three methods | Disk for a single curve, washer for a region between two curves, and shell for wrapping cylindrical shells around the axis. | Calculus problems on solids of revolution |
| Function input | Type f(x), and g(x) for the washer method. Supports polynomials, trig, inverse trig, exp, ln, log, sqrt, pi and e. | Entering the curves exactly as written in a problem |
| Bounds | Set a lower bound a and an upper bound b for the integral. | Regions with specific limits, such as 0 to 2 |
| Axis of rotation | Rotate around the x-axis, the y-axis or a custom axis value such as -1 or 3. | Problems that rotate about a line other than an axis |
| Steps and comparison | Shows the integral formulas with your numbers and compares all three methods. | Seeing why one method is easier for a given setup |
| Profile and 3D solid | Switch between a 2D profile and a 3D solid, with an Animate option. | Visualizing the shape before trusting the number |
| Preset examples | Buttons load x² disk, sin(x), √x paraboloid, x vs x² washer and x² shell examples. | Learning each method from a worked case |
Three Methods of Calculation
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 parallel 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 perpendicular 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.
Reference this content, page, or tool as:
"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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