How do i find the area of the following shaded region?
The figure consists of two circles, one of radius $2r$ and the other of radius $r$.
The distance of the center of the circle of radius $r$ from the bigger circle is $r$, as shown.
The total angle is $\theta$---$\left(\dfrac{\theta}{2}+\dfrac{\theta}{2}\right)$
If we take a Cartesian reference in the center of the small circle, so that $$ x^2+y^2=r^2\\ (x+r)^2+y^2=(2r)^2 $$ are the equations of the two circles, then in polar coordinates the two equations reduce to $$ \rho=r\\ \rho=r\left(\sqrt{3+\cos^2\phi}-\cos\phi\right) $$ The requested area is given by $$ A=\int_{-\theta/2}^{+\theta/2}d\phi\int_r^{r\left(\sqrt{3+\cos^2\phi}-\cos\phi\right)}\rho d\rho $$