Quick answer: rotating y = f(x) about the x-axis gives V = π ∫ₐᵇ [f(x)]² dx (disk method). With a hole between two curves, subtract: π ∫ (R² − r²) dx (washer). Rotating about the y-axis, shells give V = 2π ∫ x·f(x) dx. Enter your function and bounds using x — for example x^2, sqrt(x), sin(x).
The three methods
Washer (region between R and r): V = π ∫ₐᵇ ([R(x)]² − [r(x)]²) dx
Shell (about y-axis): V = 2π ∫ₐᵇ x · f(x) dx
Each method slices the solid differently. The disk method cuts perpendicular to the axis into solid circular slices; the washer method does the same when there is a hole, subtracting the inner disk; the shell method cuts into concentric cylindrical tubes. All three give the same volume for the same solid — you choose whichever produces the simpler integral.
A worked example you can check
Rotate y = x² about the x-axis from x = 0 to x = 2. By the disk method:
- V = π ∫₀² (x²)² dx = π ∫₀² x⁴ dx
- = π [x⁵/5]₀² = π × 32/5
- = 20.106…
Enter x^2 with bounds 0 and 2 in disk mode and the calculator returns the same value — a good way to confirm the tool matches the analytic answer before trusting it on a harder function.
The disk, washer and shell methods are standard results of integral calculus. This calculator evaluates the resulting definite integral numerically with a fine Simpson's-rule scheme.
Function syntax
| Write | Meaning |
|---|---|
| x^2 | x squared |
| sqrt(x) | square root of x |
| sin(x), cos(x), tan(x) | trig functions (radians) |
| exp(x), ln(x) | e^x and natural log |
| 2*x + 1 | use * for multiplication |
| pi | π ≈ 3.14159 |
Frequently asked questions
What is the disk method?
The disk method finds the volume of a solid formed by rotating a curve about an axis by slicing it into thin disks. For rotation about the x-axis, V = π ∫ [f(x)]² dx between the bounds. Each slice is a disk of radius f(x) and thickness dx, and the integral adds them up.
What is the washer method?
The washer method handles a solid with a hole through it, formed between two curves. V = π ∫ ([R(x)]² − [r(x)]²) dx, where R is the outer radius and r the inner. Each slice is a washer — a disk with a smaller disk removed from its centre.
What is the shell method?
The shell method slices the solid into thin cylindrical shells instead of disks. For rotation about the y-axis, V = 2π ∫ x·f(x) dx. It is often easier than the disk method when rotating about an axis the region does not touch, or when the function is hard to invert.
When do I use each method?
Use disks or washers when slicing perpendicular to the axis of rotation gives simple radii — washers when there is a gap between the region and the axis. Use shells when slicing parallel to the axis is simpler, which often happens when rotating about the y-axis. Both give the same answer; the choice is about which integral is easier.
How does this calculator evaluate the integral?
It evaluates the volume integral numerically using a fine Simpson's-rule approximation, which is accurate to many decimal places for smooth functions. For a symbolic answer, set up the integral by hand using the method shown and integrate analytically.