I have a question that I think may be hard, so I am trying to understand it more generally.
Say $f$ is a smooth function on the unit circle, so it is a $2\pi$ periodic function on $\mathbb{R}$. I am concerned with integrals of the form $$r\int_0^\pi f(\theta)\sin\theta d\theta,$$ where $r$ is some positive constant (the circle's radius). This can be thought of as in some sense scaling the contributions of the infinitesimal elements of the circumference in polar coordinates.
Now, my issue is that I am getting somewhat complicated expressions for $f$, and there seems to be no way to evaluate the integral. It occurred to me, however, that Fourier analysis or convolution might be able to help, or that wise people might have solved this problem generally.
My question is: am I right about suspecting a general theory or some kind of clever techniques? If so, what are they? If not, are there any tools I can use to help with problems of this sort?