Method guideCalculus IIAdvanced11 min

Surface area of revolution: radius times arc length

Build the integral from the distance to the axis and the differential arc length, then check that the chosen function stays nonnegative where required.

LaTeX article Updated July 13, 2026

S=2\pi\int_a^b r(x)\sqrt{1+[f'(x)]^2}\,dx

The radius is distance to the axis

Rotating around the x-axis uses radius |y|, while rotating around the y-axis uses radius |x|. On a nonnegative interval those absolute values may simplify.

State the axis before writing the integral. Reusing the wrong radius is the surface-area version of mixing up shell and washer geometry.

\text{x-axis: }r=|f(x)|\qquad\text{y-axis: }r=|x|

Arc length supplies the slanted width

For y = f(x), a differential curve segment has length ds = √(1 + [f′(x)]²) dx. Surface area uses that slanted segment, not merely dx.

If x is naturally a function of y, the corresponding element is √(1 + [g′(y)]²) dy. Choose the orientation that keeps both radius and derivative manageable.

ds=\sqrt{1+[f'(x)]^2}\,dx

The result has square units

Circumference has units of length and ds has units of length, so their product has area units. This dimensional check catches missing radius or arc-length factors.

Surface area integrals are often algebraically difficult. Setting up the correct exact integral is meaningful even when numerical evaluation is required.

[2\pi r\,ds]=L\cdot L=L^2

Worked example

Common mistakes

Using dx instead of the arc-length element.
Measuring radius from the wrong axis.
Reporting cubic units for a surface area.

Keep these ideas

Surface area is circumference times slanted width.
Radius means distance to the rotation axis.
The result must have square units.