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