What is the advantage of describing functions in terms of polar coordinates instead of Cartesian coordinates?

Question

Using $x = rcos(\theta)$, $y = rsin(\theta)$, we can rewrite $x^2 = y$ as $r = sin(\theta)sec^2(\theta)$

This seemed very unnecessary while I was learning calculus. Does anyone know if there are specific instances where it is advantageous of describing Cartesian functions in polar coordinates aside from drawing pretty intersecting curves?

Answer 1

many times integrating over a certain region such as a circle in cartesian coordinates would require you to set up two integrals whereas in polar coordinates the integral would be much simpler to compute.