Arc Length & Surface Area of Revolution Cheat Sheet

Key Facts

• For $y=f(x)$ on $a\le x\le b$, arc length is $L=\int_a^b \sqrt{1+\left(f'(x)\right)^2}\,dx$.
• For $x=g(y)$ on $c\le y\le d$, arc length is $L=\int_c^d \sqrt{1+\left(g'(y)\right)^2}\,dy$.
• For parametric curves $x=x(t)$ and $y=y(t)$ on $\alpha\le t\le \beta$, arc length is $L=\int_{\alpha}^{\beta}\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\,dt$.
• For polar curves $r=r(\theta)$ on $\alpha\le \theta\le \beta$, arc length is $L=\int_{\alpha}^{\beta}\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}\,d\theta$.
• Surface area of revolution is $S=\int 2\pi r\,ds$, where $r$ is the distance from the curve to the axis of rotation.
• If $y=f(x)$ rotates about the $x$-axis, then $S=2\pi\int_a^b f(x)\sqrt{1+\left(f'(x)\right)^2}\,dx$ when $f(x)\ge 0$.
• If $y=f(x)$ rotates about the $y$-axis, then $S=2\pi\int_a^b x\sqrt{1+\left(f'(x)\right)^2}\,dx$ when $x\ge 0$.
• Always check the interval and use a nonnegative radius, because surface area cannot be negative.

Vocabulary

Arc length

Arc length is the total distance along a curve between two endpoints.

Differential arc length

Differential arc length, written $ds$, represents a tiny piece of curve length used inside an integral.

Surface of revolution

A surface of revolution is formed when a curve is rotated around an axis.

Radius of rotation

The radius of rotation is the perpendicular distance from a point on the curve to the axis of rotation.

Parametric curve

A parametric curve gives coordinates as functions of a parameter, usually written $x=x(t)$ and $y=y(t)$.

Polar curve

A polar curve describes points using distance $r$ from the origin and angle $\theta$ from the positive $x$-axis.