The problem involves: mapping a Cartesian function (with domain $0$ - $2\pi$) onto the polar plane.This transformation maps the value of the Cartesian function for every angle around the center point.
E.g $f(x) = 1$ would map the unit circle
$g(x) = x$ would map a spiral
The problem is to find a generalized method of finding the length of the mapped 'curves'
For the general Cartesian function $f(x)$, its length from $a$ to $b$ is $L= \int_a^b \sqrt{1+f'(x)^2}$.
Update: Answer found for 'curved' length $L= \int_0^{2\pi} \sqrt{f(x)^2+f'(x)^2}$. I am not excactly sure why this works but arrived to this conclusion by trial and error