Hello, kittycat!
Use a double integral in polar coordinates to find the area of the region
inside the circle $\displaystyle x^2 + y^2 \:=\:4$ and to the right of the line $\displaystyle x=1$. Code:
Y

* * * 
*  *P
*  :*
*  ::*
 :::
*  :::*
  *     +   +::*  
* O 1:::*2 X
 :::
*  ::*
*  :*
*  *Q
* * * 

The polar equation of the circle is: .$\displaystyle r \,=\,2$
The polar equation of $\displaystyle x = 1$ is: .$\displaystyle r \,=\,\sec\theta$
Draw radii $\displaystyle OP$ and $\displaystyle OQ$.
We find that: .$\displaystyle P(1,\,\sqrt{3}),\;Q(1,\,\text{}\sqrt{3})$
. . and that: $\displaystyle \angle POX \,=\,\angle QOX \,=\,\frac{\pi}{3}$
The double integral is: .$\displaystyle A \;=\;\int^{\frac{\pi}{3}}_{\text{}\frac{\pi}{3}} \int^2_{\sec\theta} r\,dr\,d\theta $