- 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: $$\mathbf{r}(t) = t\mathbf{i} + 3t^2\mathbf{j} + 6t^3\mathbf{k}, \quad 0 \leq t \leq 2$$
- Velocity $\mathbf{v}(t)$ is $$ \begin{align*} \mathbf{v}(t) &= \mathbf{i} + 6t\mathbf{j} + 18t^2\mathbf{k} \end{align*} $$ Speed $v(t) = \|\mathbf{v}(t)\|$ is $$ \begin{align*} v(t) &= \sqrt{1 + 36t^2 + 18^2t^4} \\ &= \sqrt{\left(1 + 18t^2\right)^2} \\ &= 1 + 18t^2 \end{align*} $$ Arc length $s$ is $$ \begin{align*} s &= \int_0^2 v(t)\, dt \\ &= \int_0^2 1 + 18t^2\, dt \\ &= t + 6t^3\, \Biggr|_0^2 \\ &= 50 \quad \blacksquare \end{align*} $$