Arc length and surfaces of revolution

What you’ll learn

• Calculate lengths of Cartesian and parametric curves.
• Build surface-area integrals using the radius of rotation.

Understand the idea

A short curve segment behaves like the hypotenuse of a right triangle with changes $dx$ and $dy$. Integrating its length gives the arc-length formula. For a parametric curve, differentiate both coordinates with respect to the same parameter and use the corresponding parameter limits.

Rotating a short arc about an axis creates a narrow band with area approximately circumference times arc length. About the x-axis, the radius is the distance $|y|$; about the y-axis, it is $|x|$. Surface area therefore uses arc length, whereas a volume-of-revolution integral uses circular cross-sectional area.

Key formulae

$$L=\int_a^b\sqrt{1+\left(\frac{dy}{dx}\right)^2}\,dx$$

$$L=\int_\alpha^\beta\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\,dt$$

$$S_x=2\pi\int |y|\,ds$$

$$S_y=2\pi\int |x|\,ds$$

Worked example

Find the surface area made by rotating $y=x$, $0\le x\le2$, about the x-axis.

1. Here $dy/dx=1$, so $ds=\sqrt2\,dx$.
2. The radius is $y=x$, giving $S=2\pi\sqrt2\int_0^2x\,dx$.
3. The integral is $[x^2/2]_0^2=2$.

Answer: $4\pi\sqrt2$ square units.