- 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: $$\mathbf{r}(t) = t\mathbf{i} + \log(\sec t)\mathbf{j} + \log(\sec t + \tan t)\mathbf{k}, \quad 0 \leq t \leq \frac{1}{4}\pi$$
- Velocity $\mathbf{v}(t)$ is $$ \begin{align*} \mathbf{v}(t) &= \mathbf{i} + \tan t\,\mathbf{j} + \sec t \,\mathbf{k} \end{align*} $$ Speed $v(t) = \|\mathbf{v}(t)\|$ is $$ \begin{align*} v(t) &= \sqrt{1 + \tan^2 t + \sec^2 t} \\ &= \sqrt{\frac{\cos^2 t + \sin^2 t + 1}{\cos^2 t}} \\ &= \sqrt{2}\sec t \end{align*} $$ Note: From our calculation of velocity, we saw that $\frac{d}{dt}\log (\tan t + \sec t) = \sec t.$ Arc length $s$ is: $$ \begin{align*} s &= \int_0^{\pi/4} v(t)\, dt \\ &= \sqrt{2}\int_0^{\pi/4}\sec t\, dt \\ &= \sqrt{2}\log\left(\tan t + \sec t\right)\,\Biggr|_0^{\pi/4} \\ &= \sqrt{2}\log\left(1 + \sqrt{2}\right) \quad \blacksquare \end{align*} $$