Disk method
Use slices perpendicular to the axis when the rotated region reaches the axis. One boundary and the axis determine the radius.
Volume of Revolution Calculator
Calculate the volume formed when a plane region rotates around a horizontal or vertical line. Choose disks, washers, or cylindrical shells; the calculator builds the correct integral, evaluates it numerically, and graphs the region and axis—all locally in your browser.
Define the rotated region
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Volume result
Calculated solid volume
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Enter a region, axis, and bounds to calculate its volume.
The volume integral will appear here.
The result is a numerical approximation for continuous boundary functions on the selected interval.
Region and axis of rotation
Calculate a volume to draw the generating region.
Calculator methodology and limits
- Developed and reviewed by
- Starlight Robotics calculator team
- Last reviewed
- Method
- Parse the boundaries → construct cross-sectional area or shell contribution → scan the region for invalid geometry → integrate with adaptive Simpson subdivision.
- Accuracy
- The displayed estimate comes from the final numerical subdivisions. It is useful evidence of quadrature convergence, not a proof of exact rounding error.
- Geometry limits
- Boundaries must be continuous and keep their stated order. Washer boundaries must remain on one side of the axis; shell bounds must not cross the axis.
- Privacy
- Your functions, bounds, graph, copied summary, and downloaded data remain on your device.
How to use the volume of revolution calculator
- Choose Disk for solid circular slices, Washer for slices with a hole, or Shell for slices parallel to the axis.
- Select a horizontal or vertical axis. The field labels automatically switch between functions of
xand functions ofy. - Enter the boundary functions, finite interval, and the constant in the axis line
y=corx=c. - Calculate, then inspect the integral, error estimate, and graph. Copy the summary or download the sampled cross-section data if useful.
Choosing disk, washer, or shell method
Washer method
Use perpendicular slices when the region leaves a central hole. Enter the farther boundary first so R≥r≥0.
Shell method
Use slices parallel to the axis. Each thin rectangle becomes a shell with circumference 2πr, height h, and thickness dx or dy.
Shifted axes
For rotation about y=c or x=c, radii are distances from that line—not simply the function value or coordinate.
Worked examples
Disk: y=√x on [0,4]
About y=0, R=√x. Therefore V=π∫₀⁴(√x)²dx=π∫₀⁴x dx=8π≈25.13274123.
Washer: y=2 and y=x
About y=0 on [0,2], V=π∫₀²(2²−x²)dx=16π/3≈16.75516082.
Shell: y=4−x²
Above y=0, from x=0 to x=2, rotated about x=0: V=2π∫₀²x(4−x²)dx=8π.
Volume of revolution formulas
| Method | Slice direction | Formula |
|---|---|---|
| Disk | Perpendicular to the axis | V=π∫ₐᵇ[R(u)]²du |
| Washer | Perpendicular to the axis | V=π∫ₐᵇ([R(u)]²−[r(u)]²)du |
| Cylindrical shell | Parallel to the axis | V=2π∫ₐᵇ radius(u)·height(u)du |
The variable u is x for vertical slices and y for horizontal slices. Disk and washer radii are perpendicular distances to the axis; shell radius is the distance from a parallel slice to the axis.
Reference: see OpenStax Calculus Volume 1, §6.2 for disks and washers, and Calculus Volume 2, §2.3 for cylindrical shells.
Volume of revolution calculator FAQs
When should I use the disk method?
Use disks when slices perpendicular to the axis rotate into circles with no hole. Enter the boundary of the region and the axis; their distance is the radius.
What is the difference between the washer and shell methods?
Washers subtract an inner circular area from an outer circular area and use slices perpendicular to the axis. Shells use slices parallel to the axis and integrate circumference times height.
How do I rotate around y=2 or x=−1?
Select the correct axis orientation and enter 2 or −1 as the axis value. Radius calculations automatically use the distance from that shifted line.
Why was my washer setup rejected?
The inner radius may exceed the outer radius, or the two entered boundaries may lie on opposite sides of the axis. If a cross-section reaches across the axis, it is a disk rather than a washer with a hole.
Why can’t shell bounds cross the axis?
Parallel slices on opposite sides can rotate to the same radius and overlap. Directly integrating both sides may double-count the solid, so choose washers or reformulate the region on one side.
Does reversing the bounds make volume negative?
No. Volume is geometric and nonnegative. The calculator reorders reversed bounds and reports that adjustment in the result note.
Can I enter discontinuous or piecewise boundaries?
The calculator expects continuous real-valued expressions over one finite interval. Split a valid piecewise region into continuous parts and calculate each non-overlapping contribution separately.
Does my region data leave the browser?
No. Parsing, numerical integration, graphing, copying, and CSV preparation happen locally. A permalink includes only inputs you explicitly choose to place in the URL.