Use a double integral in polar coordinates to find the area

Question

So the area is just an intersection of two circles

Converting the two circles to polar coordinates, I get:

1. $r(r-2\sin\theta)=0$, and

2. $r(r-2\cos\theta)=0$

Ummm so $r =0$ and r = $2\sin\theta$ and r=$2\cos\theta$ ? are those the boundaries?

Answer 1

Here is how you advance.

$$ \int_{0}^{\pi/4}\int_{0}^{2\sin \theta}rdrd\theta + \int_{\pi/4}^{\pi/2} \int_{0}^{2\cos \theta} rdrd\theta. $$

Note: To find the point of intersection of both circles solve the two equations

1)

$$ r=2\sin \theta,\quad r=2\cos \theta. $$

2)

$$ \iint_D dxdy = \iint_D r dr d\theta. $$