# What is the arclength of the polar curve f(theta) = sin4theta-cos3theta  over theta in [pi/3,pi/2] ?

Mar 20, 2017

The arclength is approximately $0.46371$

#### Explanation:

The equation for arclength of a polar curve is:

${\int}_{a}^{b} \sqrt{{r}^{2} + {\left(\frac{\mathrm{dr}}{d \theta}\right)}^{2}} d \theta$

In this case,

${r}^{2} = {\left(\sin 4 \theta - \cos 3 \theta\right)}^{2}$

$\frac{\mathrm{dr}}{d \theta} = \frac{d}{d \theta} \left(\sin 4 \theta - \cos 3 \theta\right)$

${\left(\frac{\mathrm{dr}}{d \theta}\right)}^{2} = {\left(4 \cos 4 \theta + 3 \sin 3 \theta\right)}^{2}$

So, the arclength can be found by:

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

$\approx 0.46371$