Arc Length of Space Curves
Textbook Reference
| Primary source | OpenStax Calculus Volume 3, Section 3.3: “Arc Length and Curvature” |
| Direct link | https://openstax.org/books/calculus-volume-3/pages/3-3-arc-length-and-curvature |
| Textbook used in class | Stewart, Calculus, Section 13.3: “Arc Length and Curvature” (Examples 1, 2) |
Quick Reference
Arc length of $\mathbf{r}(t) = \langle f, g, h\rangle$ on $[a, b]$: \[ L = \int_a^b |\mathbf{r}'(t)|\,dt = \int_a^b \sqrt{[f'(t)]^2 + [g'(t)]^2 + [h'(t)]^2}\,dt. \]
Arc length function: $s(t) = \int_a^t |\mathbf{r}'(u)|\,du$. Then $ds/dt = |\mathbf{r}'(t)|$.
Key Concepts
1. Computing Arc Length
Example 1. Find the length of the helix $\mathbf{r}(t) = \langle \cos t, \sin t, t\rangle$ from $t = 0$ to $t = 2\pi$. (Stewart 13.3, Example 1.)
$\mathbf{r}'(t) = \langle -\sin t, \cos t, 1\rangle$. $|\mathbf{r}'(t)| = \sqrt{\sin^2 t + \cos^2 t + 1} = \sqrt{2}$.
\[ L = \int_0^{2\pi} \sqrt{2}\,dt = 2\sqrt{2}\pi. \]
Boxed answer: $2\sqrt{2}\pi$.
Recap. The speed $|\mathbf{r}'(t)|$ is constant for this helix, making the integral trivial. In general, $|\mathbf{r}'(t)|$ is not constant and the integral must be computed.
2. Arc Length Parameterization
A curve is parameterized by arc length when $|\mathbf{r}'(s)| = 1$ for all $s$. In this parameterization, the parameter $s$ literally measures distance along the curve. Such a parameterization simplifies the curvature formula.
To reparameterize: find $s(t) = \int_0^t |\mathbf{r}'(u)|\,du$, invert to get $t = t(s)$, then set $\tilde{\mathbf{r}}(s) = \mathbf{r}(t(s))$.
computing arc length as $\int |\mathbf{r}(t)|\,dt$. The arc length uses $|\mathbf{r}'(t)|$ (the speed), not $|\mathbf{r}(t)|$ (the distance from the origin). The formula integrates the magnitude of the velocity, not the magnitude of the position.
Common Misconceptions
the arc length of a space curve is $\int_a^b |\mathbf{r}(t)|\,dt$, the integral of the distance from the origin.
This is the height-vs-slope error. The arc length formula integrates the speed $|\mathbf{r}'(t)|$, which is the magnitude of the velocity vector, not the magnitude of the position vector. The distance from the origin $|\mathbf{r}(t)|$ measures how far the curve is from the origin at each parameter value, which has nothing to do with how long the curve is. For the helix $\mathbf{r}(t) = \langle \cos t, \sin t, t\rangle$, the distance from the origin is not constant while the speed $|\mathbf{r}'(t)| = \sqrt{2}$ is constant.
Leveled Practice
Problem 1. Find the arc length of $\mathbf{r}(t) = \langle 3t, 4\cos t, 4\sin t\rangle$ from $t = 0$ to $t = 2\pi$.
Show answer
$\mathbf{r}'(t) = \langle 3, -4\sin t, 4\cos t\rangle$. $|\mathbf{r}'(t)| = \sqrt{9 + 16\sin^2 t + 16\cos^2 t} = \sqrt{9+16} = 5$.
$L = \int_0^{2\pi} 5\,dt = 10\pi$.