# What is the arclength of r=4theta  on theta in [-pi/4,pi]?

Sep 5, 2016

$\approx 27.879$

#### Explanation:

This is an outline method. The grind of some of the work has been done by computer.

Arc length $s = \int \dot{s} \setminus \mathrm{dt}$

and $\dot{s} = \sqrt{\vec{v} \cdot \vec{v}}$

Now, for $\vec{r} = 4 \theta \setminus \hat{r}$

$\vec{v} = \dot{r} \hat{r} + r \dot{\theta} \hat{\theta}$

$= 4 \dot{\theta} \setminus \hat{r} + 4 \theta \dot{\theta} \setminus \hat{\theta}$

$= 4 \dot{\theta} \left(\hat{r} + \theta \setminus \hat{\theta}\right)$

So $\dot{s} = 4 \dot{\theta} \sqrt{1 + {\theta}^{2}}$

Arc length $s = 4 {\int}_{{t}_{1}}^{{t}_{2}} \sqrt{1 + {\theta}^{2}} \setminus \dot{\theta} \setminus \mathrm{dt}$

$= 4 {\int}_{- \frac{\pi}{4}}^{\pi} \sqrt{1 + {\theta}^{2}} \setminus d \theta$

$= 2 {\left[\theta \sqrt{{\theta}^{2} + 1} + {\sinh}^{- 1} \theta\right]}_{- \frac{\pi}{4}}^{\pi}$ computer solution. See Youtube linked here for the method

$\approx 27.879$ computer solution