Double integrals - area intersected by polar figures

Hi,

I have been trying to follow the working provided in my book to the answer to a question on this topic, and do not quite understand how they have gone about it.

I was wondering if anyone could shed some light on this for me?


Find the area of the polar figure enclosed by the circle r=2 and the cardiod $r=2(1+cos\theta)$

I made a sketch to visualise the problem, with the shaded part the area sought:



The first part of the working is what I don't understand:

$A = 2\int^{\frac{\pi}{2}}_0\int^{2(1+cos\theta)}_2rdrd\theta$


This is much more difficult for me to visualise compared with double integrals involving Cartesian co-ordinates.

I believe the first integral is taking 'slices' from the inner curve(circle) to the outer curve (cartoid), hence the limits of $r=2$ and $r = 2(1+cos\theta)$.

But I can not understand the upper limit placed on the next integral. Surely it should be $\pi$, rather than $\frac{\pi}{2}$...

Actually, the more I think about this, the more it seems to me that the area this line of working gives is that of the unshaded part of the cardoid, but I can't say I understand this well enough to be certain...

I cant see your attachement, so I attach the graphs.
For the half of shaded region 0<=\theta <=\pi/2

From pi/2 tp pi the inner curve is the cardioid and the circle is the outer.