Calculator Inputs
Use explicit multiplication like 2*x and powers like x^2.
Formula Used
For a region rotated about the x-axis, the volume of revolution is based on circular cross-sections. The outer radius is the distance from the x-axis to the outer curve, and the inner radius is the distance to the hollow boundary.
General washer formula:
V = π ∫[R(x)2 - r(x)2] dx
Disk formula:
V = π ∫R(x)2 dx when r(x) = 0.
Numerical method used on this page:
This calculator applies Simpson’s Rule to estimate the definite integral. It divides the interval into even subintervals, evaluates the squared radii, weights each sample, and multiplies the final integral by π.
How to Use This Calculator
- Enter the outer function using
xas the variable. - Enter an inner function only if the rotated solid has a hollow center.
- Provide lower and upper x-bounds for the interval of rotation.
- Choose the disk or washer method as needed.
- Set the interval count, precision, graph points, and a unit label.
- Press Calculate Volume to view results, graph, CSV export, and PDF export.
Example Data Table
| Case | Outer Function | Inner Function | Bounds | Method | Approximate Volume |
|---|---|---|---|---|---|
| Example 1 | x^2 | 0 | 0 to 2 | Disk | 20.1062 units3 |
| Example 2 | 3 | 1 | 0 to 4 | Washer | 100.5310 units3 |
| Example 3 | sqrt(x) | 0 | 0 to 9 | Disk | 127.2345 units3 |
Frequently Asked Questions
1) What formula does this calculator use?
A solid rotated about the x-axis uses V = π∫[R(x)2 - r(x)2]dx. For a full solid, the inner radius is zero. The page evaluates this numerically with Simpson’s Rule across your chosen interval count.
2) When should I use disk instead of washer?
Choose disk when the region touches the x-axis, so no hole exists. Choose washer when a second boundary creates an inner hollow radius. Both use the same base idea, but washer subtracts the inner contribution.
3) Why are radii treated as distances?
Rotation radius is the distance from the x-axis, not the signed y-value. Using distance keeps the cross-sectional area positive even when the curve falls below the axis.
4) Why does the calculator force an even interval count?
Simpson’s Rule works with paired subintervals. An even count lets the algorithm combine parabolic slices correctly and usually improves accuracy compared with very coarse estimates.
5) Is the result exact?
No. This page gives a numerical approximation unless the function fits the sampling perfectly. Increasing intervals usually improves stability and reduces approximation error.
6) What functions can I type?
You can enter x, numbers, parentheses, +, -, *, /, ^, and functions such as sin, cos, tan, sqrt, abs, log, ln, exp, pow, sec, csc, and cot. Use explicit multiplication like 2*x.
7) What does the graph show?
The chart plots the outer function, optional inner function, and cross-sectional area trend over the chosen x-range. It helps you see where the radius grows, shrinks, or changes rapidly.
8) Can I use centimeters, meters, or inches?
Yes. Enter any consistent unit label. If inputs are in centimeters, the result is reported in cubic centimeters. Consistent units matter more than the specific label you choose.