Math help, polar curve area between the curves

Question

anonymous · Answer

Find the area inside the circle $\large{r=2}$, but outside the cardiod $\large{r=2-2\cos\theta}$. And here is the sketch. https://www.desmos.com/calculator/azsqobis1k

anonymous · Answer

I need help with the set up. pliz

irishboy123 · Answer

i'd do it in the first quad and rely upon symmetry.  the circle is easy.  the cardiod goes from theta = 0 to π/2.

phi · Answer

inner integral  over r from 2-2cos theta  to 2
outer integral,  we need the range for theta

anonymous · Answer

Sorry, but what do you refer to by an inner integral?

phi · Answer

$$ \int \int_{2-\cos \theta}^2 \ r \ dr \ d\theta $$

anonymous · Answer

double integral?

anonymous · Answer

we haven't learned anything like this in class.
We have learned the basic formula for polar curves   $\large{\displaystyle A=\int_{a}^{b}(1/2)~r^2~d\theta}$, and have previously done areas between Cartesian curves.

This however, is a little new.

phi · Answer

ok.  it's clear we want to find the area of a sector of the circle and subtract off the cartiod

anonymous · Answer

yeah that seems logical.

anonymous · Answer

I thought I should solve for intersection points (sorry if I am rushing)

$\large{2=2-2\cos\theta}$

$\large{0=-2\cos\theta}$

$\large{0=\cos\theta}$

$\large{\theta=\pi/2,~~3\pi/2}$

phi · Answer

we could try to get tricky, but I would find the area of
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