Length of a Polar Curve

1. Sep 8, 2010

Exeneva

1. The problem statement, all variables and given/known data
r = 3 sin $$\vartheta$$

0 $$\leq$$ $$\vartheta$$ $$\leq$$ $$\pi$$/3

2. Relevant equations

Arc Length: $$\int$$ $$\sqrt{r^{2} + (dr/d\vartheta)^{2}}d\vartheta$$

3. The attempt at a solution
$$r^{2} = 9 (sin \vartheta)^{2} = 9 (1/2 - cos 2\vartheta/2)$$

$$r^{2} = 9/2 - 9/2 cos 2\vartheta$$

$$dr/d\vartheta = 3 cos \vartheta$$

$$\int (9/2 - 9/2 cos 2\vartheta + 3 cos \vartheta)^{1/2} d\vartheta$$
from 0 to \pi/3

Not sure how to integrate this.

2. Sep 8, 2010

Staff: Mentor

You really made it hard on yourself.

$$r^{2} = 9 sin^2( \theta)$$
$$(dr/d\theta)^2 = (3 cos(\theta))^{2} = 9cos^2(\theta)$$