r = 2\,\text{ is a circle with center at }(0,0)\text{ and radius 2.}
r =4\sin\theta\,\text{ is a circle, center at }(0,2)\text{ and radius 2.}
. . [I'm speaking in rectangular coordinates, obviously.]
The circles intersect at \theta = \tfrac{\pi}{6},\:\tfrac{5\pi}{6}
Due to the symmetry, we can find the area of the right half
. . then multiply by 2.The area of the right half is: