- 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: $$\mathbf{r}(t) = a(\sinh t - t)\mathbf{i} + a(\cosh t - 1)\mathbf{j}, \quad 0 \leq t \leq T, \quad a > 0$$
- Velocity $\mathbf{v}(t)$ is $$ \begin{align*} a\left(\cosh t -1\right) \mathbf{i} + a\left(\sinh t\right)\mathbf{j} \end{align*} $$ Speed $v(t) = \|\mathbf{v}(t)\|$ is $$ \begin{align*} a\left(\cosh^2 t + \sinh^2 t + 1 - 2 \cosh t\right)^{1/2} \end{align*} $$ Arc length $s$ is $$ \begin{align*} s &= \int_0^{T}v(t)\,dt \\ &= \int_0^T a\left(\cosh^2 t + \sinh^2 t + 1 - 2 \cosh t\right)^{1/2}\, dt \end{align*} $$