Area between two polar curves using iterated integrals?

Question

The question is from a practice exam I am currently trying to do:

I am really not sure how to go about this one. In essence, I'd imagine that the idea is to find the area of the greater curve, and then from that subtract the calculated area of the smaller one. How do I go about doing this? I figure you can do this with one iterated integral?

Further, if possible, as an aside to this question, I am also wondering how to find the area of just one of these curves with iterated integrals?

Thank you very much!

Answer 1

You can just calculate the area of the $\text{outer curve} - \text{inner curve}$:

$$\frac{1}{2} \int_0^{2\pi} \big(4 - \cos(3\theta) \big)^2 - \big(2 + \sin(3 \theta) \big)^2$$

Answer 2

$$\int\limits_{\theta = 0}^{2 \pi} \int\limits_{r = 2 + \sin (3 \theta)}^{4 - \cos (3 \theta)} r\ dr\ d\theta = 12 \pi$$