# How do you find the exact length of the polar curve r=3sin(theta) on the interval 0<=theta<=pi/3 ?

Sep 21, 2014

The arc length is $\pi$.

Let us look at some details.

$r = 3 \sin \theta$

by differentiating with respect to $\theta$,

$R i g h t a r r o w \frac{\mathrm{dr}}{d \theta} = 3 \cos \theta$

So, the arc length L can be found by

$L = {\int}_{0}^{\frac{\pi}{3}} \sqrt{{r}^{2} + {\left(\frac{\mathrm{dr}}{d \theta}\right)}^{2}} d \theta$

$= {\int}_{0}^{\frac{\pi}{3}} \sqrt{{3}^{2} {\sin}^{2} \theta + {3}^{2} {\cos}^{2} \theta} d \theta$

by pulling $3$ out of the square-root,

$= 3 {\int}_{0}^{\frac{\pi}{3}} \sqrt{{\sin}^{2} \theta + {\cos}^{2} \theta} d \theta$

by ${\sin}^{2} \theta + {\cos}^{2} \theta = 1$,

$= 3 {\int}_{0}^{\frac{\pi}{3}} d \theta = 3 {\left[\theta\right]}_{0}^{\frac{\pi}{3}} = 3 \left(\frac{\pi}{3} - 0\right) = \pi$