Area in Polar Coordinates

Question

The graph of $$r=24\cos(\theta)-12$$ is a limacon. The region between its inner and outer loops is shaded in the figure. That shaded region has area ____________ square units.

I don't know how to approach this question I have for homework. I would really appreciate the help in solving this question.

Answer 1

Hint. Your basic area formula in polar coordinates is $$A=\frac{1}{2}\int_\alpha^\beta r^2\,d\theta\ .$$ You need to do the following.

1. Find the $\theta$ values which give you the "outer" part of the curve. (Hint: what $\theta$ values give $r=0$?)
2. Apply the above integration formula. This will give the area inside the outer curve, that is, the shaded area plus the little unshaded area.
3. Repeat (1) and (2) for the "inner" part of the curve. This will give the little unshaded area.
4. Subtract (3) from (2).

Good luck!