# How to find the area of the shaded region? [closed]

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 $\theta$---$\left(\dfrac{\theta}{2}+\dfrac{\theta}{2}\right)$ ## closed as off-topic by pkwssis, Gina, Adam Hughes, Cookie, WilliamJul 25 '14 at 3:59

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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 $$A=\int_{-\theta/2}^{+\theta/2}d\phi\int_r^{r\left(\sqrt{3+\cos^2\phi}-\cos\phi\right)}\rho d\rho$$
Represent the shaded region as an algebraic sum ($+$- and $-$- signs) of circular sectors and triangles.