Arc length of a space curve

The length of a curve traced by $\mathbf{r}(t)$ , found by integrating speed over time.

The arc length of a space curve is $L=\int_a^b|\mathbf{r}'(t)|\,dt$ — integrating speed over time, the same "distance equals the integral of speed" idea from single-variable motion problems, now with three components' worth of speed bundled under one square root: $|\mathbf{r}'(t)|=\sqrt{[f'(t)]^2+[g'(t)]^2+[h'(t)]^2}$ . The smoothness requirement $\mathbf{r}'(t)\ne\mathbf{0}$ rules out a curve that stops and instantaneously reverses.

Examples

The helix $\mathbf{r}(t) = \langle 3\cos t, 3\sin t, 4t\rangle$ for $t\in[0,2\pi]$ has length $L=\int_0^{2\pi}5\,dt=10\pi$ .