- Calculus, Volume 1: One Variable Calculus, with an Introduction to Linear Algebra
- Tom M. Apostol
- Second Edition
- 1967
- 978-1-119-49673-1
14.13 Exercises
- Find the length of the path traced out by a particle moving on a curve according to the given equation during the time interval specified: $$\mathbf{r}(t) = \sin t\mathbf{i} + t\mathbf{j} + (1 - \cos t)\mathbf{k}, \quad 0 \leq t \leq 2\pi$$
- Velocity $\mathbf{v}(t)$ is $$ \begin{align*} \mathbf{v}(t) &= \cos t\,\mathbf{i} + \mathbf{j} + \sin t \,\mathbf{k} \end{align*} $$ Speed $v(t) = \|\mathbf{v}(t)\|$ is $$ \begin{align*} v(t) &= \sqrt{\cos^2 t + \sin^2 t + 1} \\ &= \sqrt{2} \end{align*} $$ Arc length $s$ is $$ \begin{align*} s &= \sqrt{2}\int_0^{2\pi}\,dt \\ &= \sqrt{2}\,t\,\Biggr|_0^{2\pi} \\ &= 2\sqrt{2}\,\pi \quad \blacksquare \end{align*} $$