How to change order of integration?

Question

$$\int_0^{\frac{\pi}{2}}\int_0^{2a \cos \theta} f(r, \theta) \ dr \ d\theta$$

Hi everyone, Can you please help me in changing the order of integration in polar coordinates?

A general theory or method will be appreciated. I had searched a lot but unable to find the related theory.

Thanks

Answer 1

Here is a slightly different way to do it

The region is a semicircle centered at (a,0)

i.e.

$x^2 + y^2 = 2ax\\ x = r\cos \theta\\y=r\sin\theta\\ r^2 = 2ar\cos \theta\\ r = 2a\cos\theta$

$u = r-a\\ \phi = 2\theta$

$\int_0^a\int_0^\pi f(u+a, \frac {\phi}{2}) (2\ d\phi \ du)$

Not exactly what you were asked for.