Disk and washer volumes
Build a solid of revolution from circular cross-sections.
Disk and washer volumes is one of 7 applications formulas in the calculus section of this library, and it is used at ap · university level.
Why disk and washer volumes works
Slice the solid at right angles to the axis it was spun around and every slice is a flat circle, possibly with a circular bite taken out of the middle. The face area of a slice is π times the outer radius squared minus π times the inner radius squared, and the integral stacks all those slices together.
What each symbol means
$R$ is outer radius and $r$ inner radius.
Disk and washer volumes: when it holds
$R\ge r\ge0$ and radii are perpendicular distances to the axis of rotation.
When it stops applying
The slice has to be a complete circle, so the cut must be perpendicular to the rotation axis and the region must reach that axis for a plain disk. When a gap separates the region from the axis you need the washer version, and when slices parallel to the axis are more natural, shells do the job better.
Disk and washer volumes: a worked example
Rotating $y=x$ on $[0,1]$ about the $x$-axis gives $V=\pi\int_0^1x^2dx=\pi/3$.
The mistake to avoid
What people do: Squaring the difference of the two radii instead of subtracting the two squares.
Why it goes wrong: Those are not the same quantity, and the difference-squared version removes far too much, so the reported volume is much too small.
Do this instead: Square each radius on its own, subtract, and then integrate the result.
Disk and washer volumes: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $R$ is outer radius and $r$ inner radius.
- Check the conditions before substituting. $R\ge r\ge0$ and radii are perpendicular distances to the axis of rotation.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most calculus slips.
Where this formula fits
- Subject
- Calculus formulas — 42 entries in this library
- Topic
- Applications
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where disk and washer volumes comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Fundamental Theorem of Calculus — the lesson behind this formula: connect accumulated area with instantaneous change.
- Calculus Calculator — check your substitution and the value it produces.
- Study calculus — the subject guide that explains the ideas these formulas compress.
- Calculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 42 calculus formulas — the full grouped reference, or the complete formula library.
Questions about disk and washer volumes
How do I tell a disk problem from a washer problem?
Check whether the region touches the axis of rotation. If it does there is no hole and the inner radius is zero; a gap means a hole and a washer.
What if I rotate about a line such as y = 2?
Radii are measured from that line rather than from an axis, so each radius becomes a distance to y = 2. The rest of the setup is unchanged.
Why does this use π and not 2π?
Each slice is the face of a circle, whose area carries a single π. The 2π belongs to the shell method, where a slice is unrolled into a rectangle.
Should I integrate in x or in y?
Match the variable to the direction of slicing. Spinning around a horizontal axis gives slices of small width in x, and a vertical axis gives slices of small width in y.