Areas And Lengths In Polar Coordinates

(requires JavaScript)

1. Find the area of the region that is bounded by the curve $r=\sin <!--FUNCTION\; APPLICATION-->\left(\theta \right)$ and within the sector $\pi /3\le \theta \le 2\pi /3$.

   ${\displaystyle \frac{\pi }{12}+\frac{\sqrt{3}}{8}}$

2. Find the area of the shaded region if $r=4+3\sin <!--FUNCTION\; APPLICATION-->\left(\theta \right)$.

   

   ${\displaystyle \frac{41\pi }{4}}$
3. Sketch the curve $r=2\sin <!--FUNCTION\; APPLICATION-->\left(3\theta \right)$ and find the area enclosed by it.
   $\pi$
4. Find the area of the region enclosed by a single inner loop of the curve $r=1+2\sin <!--FUNCTION\; APPLICATION-->\left(\theta \right)$.
   ${\displaystyle \pi -\frac{3\sqrt{3}}{2}}$
5. Find the area of the region which lies inside both $r=\sin <!--FUNCTION\; APPLICATION-->\left(\theta \right)$ and $r=\cos <!--FUNCTION\; APPLICATION-->\left(\theta \right)$.
   ${\displaystyle \frac{\pi -2}{8}}$

6. Find the area of the shaded region if the curves you see are $r=e^{\theta /2}$ and $r=\theta$.

   

   ${\displaystyle \frac{e^{\pi /2}}{2}-\frac{1}{2}-\frac{{\pi }^3}{48}\approx 1.26}$.