How to use this calculator
Enter the curve farther from the axis as f(x) and the nearer one as g(x); leave g = 0 when the region reaches the axis (the disk method). Give the x-interval and choose the axis. A horizontal axis y = c uses disks or washers; a vertical axis x = c uses cylindrical shells. The large number is the volume, and when it is a simple multiple of π that form is shown too.
Formulas
Worked example
y = √x from 0 to 4 about the x-axis (the default)
- Each slice is a disk of radius √x, so its area is π(√x)² = πx.
- V = π ∫04 x dx = π [x²/2]04 = π × 8.
- V = 8π ≈ 25.13 cubic units.
Disks or shells?
Both give the same volume. Slices perpendicular to the axis are disks or washers; slices parallel to the axis are shells. Pick the one whose integral is easier: with a vertical axis and the curves written as y = f(x), shells avoid solving for x in terms of y.
Frequently asked questions
Why is my answer different from the book?
Check that the right curve is outer: the outer radius is the distance from the axis to the farther curve. Also check the axis: rotating about y = −1 instead of the x-axis changes every radius by 1.
What if the region crosses the axis?
Then the two sides sweep through the same space. Split the region at the axis and use the larger radius on each part; the calculator warns you when this happens.