A fairly ubiquitous object in elementary calculus is a function of the shape $r = f(\theta)$, where $r$ is the radius and $\theta$ the argument. Common examples include the cardiod and limacon, and of course the circle can also be expressed this way.

I understand the value of the substitution $x = r\cos(\theta), y = r\sin(\theta)$ since this helps with understanding complex variables, and when doing contour integration helps immensely. But I don't recall in my entire undergraduate career, did I ever encounter a function of the shape $r = f(\theta)$ past the examples shown in first year calculus.

Are these objects of any importance to any mathematically related field? If so, what are they? If not, then why are these objects still so dominant in the curriculum?

I understand one 'application' is arc length computation, which in Cartesian coordinates is notoriously difficult (to do exactly). For instance, the arc length of $y= \sin(x)$ say is 'difficult' since $\sqrt{1 + \cos^2(x)}$ has no anti-derivative in elementary functions. If we have $r = f(\theta)$ then the integral to be computed looks something like $\displaystyle \int_{\alpha}^\beta f(\theta)d\theta$, which is fairly straightforward. But this 'application' doesn't really simplify things in practice, since a relationship of the form $r = f(\theta)$ itself is rare and difficult to obtain, and numerical methods can give us practical ways to compute arc length anyway.

Any insight would be appreciated.