Let $f$ be a positive continuous function of $2 \pi$. Express the area inside the polar curve $r = f(\theta)$ in terms of the Fourier coefficients for $f$.
We know from freshman calculus that the area under a polar curve is expressed as $A= \int_{0}^{2\pi}\frac{1}{2}r^2d\theta$. Furthermore, the Fourier coefficients for $f$ will look like $c_n = \frac{1}{2\pi} \int_{0}^{2\pi}re^{-in\theta}d\theta$. So I want to express $A$ in terms of $c_n$. However, I am having a hard time handling the complex exponential as I have very little experience manipulating these sort of objects. Is it simple property manipulation of $e^{-in\theta}$ or is there something deeper that I should be considering here?