About the Volume of Revolution Calculator
This volume of revolution calculator finds the volume of the solid you get by rotating a region of the plane around an axis. Enter the curve y = f(x), an optional second curve y = g(x) that bounds the region (0 for the x-axis), the interval [a, b] and the axis. Choose the disk/washer method to rotate about a horizontal line y = k, or the shell method to rotate about a vertical line x = k such as the y-axis.
It is designed for calculus II students checking solids-of-revolution homework, and for engineers or makers estimating the volume of a turned part, vase or tank profile. Alongside the volume you get the volume as a multiple of π (handy for matching exact answers like 8π), the area of the region, the integral that was set up and a table of radii along the interval.
The integral is evaluated numerically with Simpson's rule on 4,000 sub-intervals, which is accurate to many digits for smooth curves. The region should lie on one side of the axis; if it crosses the axis the tool warns you, because the textbook formulas then need splitting.
With the default inputs, the volume is 25.132741 cubic units. Change any value above to recalculate instantly.
How to use the volume of revolution calculator
- 1Pick the disk/washer method for a horizontal axis, or the shell method for a vertical axis.
- 2Enter the curve f(x) and the second boundary g(x) (0 for the x-axis).
- 3Enter the interval from a to b.
- 4Set k, the axis position (0 for the x-axis or y-axis).
- 5Read the volume, its multiple of π and the integral that was set up.
Formula and method
Disk and washer method: slice the solid perpendicular to the axis. Each slice is a disk of radius R = |f(x) − k| or, if there is a hole, a washer with outer radius R and inner radius r = |g(x) − k|. Its area is π(R² − r²), and adding up the slices with an integral gives the volume.
Shell method: slice the region into thin vertical strips parallel to the axis x = k. Rotating a strip makes a cylindrical shell of radius |x − k|, height |f(x) − g(x)| and thickness dx, whose volume is 2π·radius·height·dx. Shells are usually easier when rotating about a vertical axis with y given as a function of x. Both integrals are evaluated numerically with composite Simpson's rule.
- R(x), r(x)
- Outer and inner radii, measured from the axis y = k
- f(x), g(x)
- Curves bounding the region
- k
- Position of the axis of rotation
- a, b
- x-interval of the region
Worked examples
y = √x on [0, 4] about the x-axis (disk)
Each disk has radius √x, so V = π ∫₀⁴ x dx = π · 16/2 = 8π ≈ 25.1327. The flat region under the curve has area 16/3 ≈ 5.3333.
Washer: between y = x and y = x² on [0, 1]
The outer radius is x and the inner radius is x², so V = π ∫₀¹ (x² − x⁴) dx = π(1/3 − 1/5) = 2π/15 ≈ 0.4189.
Shell: y = x² on [0, 2] about the y-axis
A shell at x has radius x and height x², so V = 2π ∫₀² x³ dx = 2π · 4 = 8π ≈ 25.1327.
Sphere of radius 2
Rotating the upper half-circle y = √(4 − x²) about the x-axis gives a sphere: V = π ∫₋₂² (4 − x²) dx = 32π/3 ≈ 33.5103, matching (4/3)πr³.
Washer about y = −1
Rotating the triangle under y = x about y = −1 leaves a hole of radius 1: V = π ∫₀¹ [(x + 1)² − 1²] dx = π(1/3 + 1) = 4π/3 ≈ 4.1888.
Frequently asked questions
When should I use the shell method instead of the disk method?+
Use disks or washers when slices perpendicular to the axis are easy to describe, and shells when it is easier to slice parallel to the axis. Rotating y = f(x) about a vertical axis (like the y-axis) is usually simplest with shells, since you avoid solving for x in terms of y.
What is the difference between the disk and washer methods?+
The disk method applies when the region touches the axis, so each slice is a solid disk of area πR². If there is a gap between the region and the axis, each slice has a hole and becomes a washer with area π(R² − r²).
How do I rotate around a line other than the axes?+
Measure radii from that line. For a horizontal axis y = k the radius is |f(x) − k|; for a vertical axis x = k the shell radius is |x − k|. Enter k in the axis position field.
Why is the answer often a multiple of π?+
Every slice is circular, so π factors out of the integral. Textbooks usually give exact answers like 8π or 2π/15; the "Volume ÷ π" output gives that coefficient directly.
Can I find the volume of a solid rotated about the y-axis using disks?+
Yes, but you must write x as a function of y and integrate in y. This calculator takes curves in x, so for a y-axis rotation choose the shell method, which gives the same volume.