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.
The disk method
Use the disk method when the region being rotated sits flush against the axis of rotation, leaving a solid with no hole through the middle.
About the y-axis: V = π ∫𝑐ᵈ [g(y)]² dy
Picture slicing the solid into thin coins perpendicular to the axis. Each coin is a circle of radius f(x) and thickness dx, so its volume is π[f(x)]² dx. Integrating adds them all up. The squaring is what people forget — it is πr², not πr.
Choose disk when: the region touches the axis along its whole length, and the axis is perpendicular to the slices you would naturally draw.
The washer method
The washer method is the disk method with a bite taken out. Use it whenever the rotated region is separated from the axis, so the resulting solid has a hollow core.
R(x) = distance from axis to the outer curve
r(x) = distance from axis to the inner curve
Each slice is now an annulus — a washer — of outer radius R and inner radius r. Its area is πR² − πr², and the volume follows.
The single most common washer error: writing (R − r)² instead of R² − r². Those are not the same thing. With R = 3 and r = 1, R² − r² = 9 − 1 = 8, while (R − r)² = 2² = 4. Subtract the squares, never square the difference.
Choose washer when: there is a gap between the region and the axis, or the region lies between two curves.
The shell method
The shell method slices the solid the other way — into nested cylindrical tubes rather than flat coins.
About the x-axis: V = 2π ∫𝑐ᵈ y · g(y) dy
Unroll one thin shell and it becomes a flat rectangle: height f(x), width 2πx (the circumference at radius x), thickness dx. That is where the 2π comes from — it is a circumference, not an area.
Choose shell when: you are rotating about the y-axis but your function is written as y = f(x), and solving for x would be awkward or impossible. Rotating y = sin(x) about the y-axis is the classic case — inverting it is a mess, but the shell integral 2π∫x·sin(x) dx is straightforward.
Which method should you use?
All three give the same answer for the same solid, so the choice is purely about which integral is easier to set up and evaluate.
| Situation | Method | Integral |
|---|---|---|
| Region touches the axis, no hole | Disk | π ∫ [f(x)]² dx |
| Gap between region and axis | Washer | π ∫ (R² − r²) dx |
| Region between two curves | Washer | π ∫ (R² − r²) dx |
| y = f(x), rotating about the y-axis | Shell | 2π ∫ x·f(x) dx |
| Function is hard to invert | Shell | 2π ∫ x·f(x) dx |
| Slices are parallel to the axis | Shell | 2π ∫ x·f(x) dx |
The quickest test: draw one representative slice. If it is perpendicular to the axis of rotation, you are using disks or washers. If it is parallel to the axis, you are using shells. Getting that orientation right settles the method before you write a single 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.