- 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: $$\mathbf{r}(t) = a(1 - \cos t)\mathbf{i} + a(t - \sin t)\mathbf{j}, \quad 0 \leq t \leq 2\pi, \quad a > 0$$
- Velocity $\mathbf{v}(t)$ is $$ \begin{align*} \mathbf{v}(t) &= a \sin t\,\mathbf{i} + a(1 - \cos t)\,\mathbf{j} \end{align*} $$ Speed $v(t) = \|\mathbf{v}(t)\|$ is $$ \begin{align*} v(t) &= [a^2 \sin^2 t + a^2(1 - \cos t)^2]^{1/2} \\ &= [a^2 (\sin^2 t + 1 - 2\cos t + \cos^2 t]^{1/2} \\ &= [a^2 (2 - 2\cos t)]^{1/2} \\ &= \left[a^2 \left(2 - 2 + 4\sin^2 \frac{t}{2}\right)\right]^{1/2} \\ &= 2a \sin \frac{t}{2} \end{align*} $$ Arc length $s$ is $$ \begin{align*} s &= \int_0^{2\pi}v(t)\,dt \\ &= 2a \int_0^{2\pi}\sin \frac{t}{2}\,dt \\ &= -4a \cos \frac{t}{2}\Biggr|_0^{2\pi} \\ &= 8a \quad \blacksquare \end{align*} $$