I need to find the mass of the area delimited by circle $r = \cos{t}$ and $r = \cos{t} + \sqrt{3} \sin{t}$.
The above is a picture attached that explains the problem. I just have a hard time finding the bounds of this. I would integrate it in polar coordinates, with $\theta$ varying from $0$ to $\pi$ and $r$ varying from $\cos{t} + \sqrt3 \sin t$ to $\cos{t}$. Does this make sense to you?
When you set the equations equal, that is $cost=cost+\sqrt{3}sint$, you find essentially two solutions, $0,\pi$ that both refer to the point $(1,0)$. The problem is that the other point of intersection (Origin) is an elusive point. That is, the curves go through the Origin for different $t$ values. A little work shows that the first curve hits the Origin for $\pi/2$, but the second curve hits it for $-\pi/6$ (Add $2\pi$ if needed). So when you set up your integral for the region between the polars, this is what you need to take into account. The area is set up by subtracting the integral from the inner curve (you know the bounds) from the integral of the outer curve (you know those bounds too now). From here, give it a try and share with us if you want.