- 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: $$\mathbf{r}(t) = a\cos \omega t\mathbf{i} + a\sin \omega t\mathbf{j} + b\omega t\mathbf{k}, \quad t_0 \leq t \leq t_1$$
- Velocity $\mathbf{v}(t)$ is $$ \begin{align*} \mathbf{v}(t) &= -a \omega \sin \omega t\,\mathbf{i} + a \omega \cos \omega t \,\mathbf{j} + b\omega \mathbf{k} \end{align*} $$ Speed $v(t) = \|\mathbf{v}(t)\|$ is $$ \begin{align*} v(t) &= \sqrt{a^2\omega^2 + b^2\omega^2} \\ &= |\omega|\sqrt{a^2 + b^2} \end{align*} $$ Arc length $s$ is $$ \begin{align*} s &= \int_{t_0}^{t_1}v(t)\,dt \\ &= |\omega|\sqrt{a^2 + b^2}\int_{t_0}^{t_1}\,dt \\ &= |\omega|\sqrt{a^2 + b^2}\,(t_1 - t_0) \quad \blacksquare \end{align*} $$