- 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: $$\mathbf{r}(t) = e^t\cos t\mathbf{i} + e^t\sin t\mathbf{j}, \quad 0 \leq t \leq 2$$
- Velocity $\mathbf{v}(t)$ is $$ \begin{align*} \mathbf{v}(t) &= \left(e^t \cos t - e^t \sin t\right)\mathbf{i} + \left(e^t \sin t + e^t \cos t\right)\mathbf{j} \end{align*} $$ Speed $v(t) = \|\mathbf{v}(t)\|$ is $$ \begin{align*} v(t) &= \left[ e^{2t}\left(\cos^2 t -\sin t \cos t + \sin^2 t\right) + e^{2t}\left(\sin^2 t +\sin t \cos t + \cos^2 t\right) \right] \\ &= \left(2e^{2t}\right) \\ &= \sqrt{2}e^t \end{align*} $$ Arc length $s$ is $$ \begin{align*} s &= \int_0^{2}v(t)\,dt \\ &= \sqrt{2}\int_0^2 e^t\,dt \\ &= \sqrt{2}e^t\,\Biggr|_0^2 \\ &= \sqrt{2}\left(e^2 - 1\right) \quad \blacksquare \end{align*} $$