Q:

Area bounded under curves

Question says:
By changing to polar coordinates,find the area bounded by the curves $x^2$+$y^2$=$2y$
,$x^2$+$y^2$=$4y$
,$x=y$ and $x=0$.

A:

In polar coordinate $x^2+y^2=r^2$, $x=r\cos{\theta}$, $y=r\sin{\theta}$ all your curves
will be rewritten as follows:
$$
r=2\sin{\theta}\\
r=4\sin{\theta}\\
\theta=\pi/4;\\
\theta=\pi/2
$$
I believe it is not a problem now to calculate an integral $\int rdrd\theta$ over 
$$
\int_{\pi/4}^{\pi/2} d\theta \int_{2\sin{\theta}}^{4\sin{\theta}}rdr
$$  

