What is the arc length of the polar curve f(theta) = 2costheta-theta  over theta in [pi/8, pi/3] ?

Jan 10, 2018

$\frac{21 \pi}{8} - \frac{\sqrt{3}}{2} - 2 + \frac{1}{\sqrt{2}} + 4 \cos \frac{\pi}{8}$

Explanation:

The given function is $r = 2 \cos \theta - \theta$

$\frac{\mathrm{dr}}{d \theta}$ would be $- 2 \sin \theta - 1$

Arc length formula is ${\int}_{a}^{b} \frac{1}{2} {r}^{2} d \theta$

The required arc length would be $\frac{1}{2} {\int}_{\frac{\pi}{8}}^{\frac{\pi}{3}} \left(4 {\sin}^{2} \theta + 4 \sin \theta + 1\right) d \theta$

=$\frac{1}{2} {\int}_{\frac{\pi}{8}}^{\frac{\pi}{3}} \left(2 - 2 \cos 2 \theta + 4 \sin \theta + 1\right)$

= [3 theta - sin 2 theta- 4 cos theta]_(pi/8)^(pi/3)]

=$3 \pi - \sin \left(\frac{2 \pi}{3}\right) - 4 \cos \left(\frac{\pi}{3}\right) - \frac{3 \pi}{8} + \sin \left(\frac{\pi}{4}\right) + 4 \cos \left(\frac{\pi}{8}\right)$

=$\frac{21 \pi}{8} - \frac{\sqrt{3}}{2} - 2 + \frac{1}{\sqrt{2}} + 4 \cos \frac{\pi}{8}$

Further calculations may be made as desired.