Surface area of revolution
Add the curved bands formed by revolving a graph around an axis.
Surface area of revolution is one of 7 applications formulas in the calculus section of this library, and it is used at ap · university level.
Why surface area of revolution works
A short piece of the curve sweeps out a narrow band when it is spun around the axis. That band is a strip whose length is the circle the piece travels, 2π times its distance from the axis, and whose width is the length of the curve piece itself. Adding every band gives the total surface.
What each symbol means
$r(x)$ is nonnegative distance from the curve to the rotation axis.
Surface area of revolution: when it holds
The curve must be sufficiently smooth; choose radius and variable to match the axis and slicing direction.
When it stops applying
The radius must be a genuine nonnegative distance to the axis. If the curve crosses the axis of rotation, that radius changes sign and the integral starts subtracting surface, so split at the crossing and use the absolute distance on each piece.
Surface area of revolution: a worked example
About the $x$-axis for $y=f(x)\ge0$, use $r(x)=f(x)$.
The mistake to avoid
What people do: Using the horizontal step as the width of each band instead of the slanted length of the curve piece.
Why it goes wrong: A tilted piece of curve is longer than its shadow on the axis, so every band is measured too narrow and the total surface comes out too small.
Do this instead: Keep the arc-length factor with the square root in the integrand, since it is what measures the true width of each band.
Surface area of revolution: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $r(x)$ is nonnegative distance from the curve to the rotation axis.
- Check the conditions before substituting. The curve must be sufficiently smooth; choose radius and variable to match the axis and slicing direction.
- 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 surface area of revolution 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 surface area of revolution
Why is the circumference used and not the area of a circle?
Each band is a thin ring rolled flat, so its length is once around the circle. A circle's face area would describe a solid disk, which is not what a curve sweeps out.
Does this include the flat ends of the solid?
No. It gives only the curved surface, so any caps that close the solid off have to be added on separately.
What is the radius when spinning around the vertical axis?
It is the horizontal distance from the curve to that axis, and the arc-length factor stays written in whichever variable you are integrating.
How can a solid have finite volume but infinite surface?
Volume weights the radius twice while surface weights it once, so a shape that narrows fast enough, like Gabriel's horn, can converge for one and diverge for the other.