> But I read a description of calculus once that used the example of trying to figure out the length of a coastline and described how one could use smaller and smaller sections to estimate the length and this is what calculus is all about.

According to https://en.wikipedia.org/wiki/Outline_of_calculus:

> Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series.

The coastline paradox is definitely an example of something that interests calculus people, and to tackle the problem they most likely will use a bunch of concepts and theorems of calculus.

However calculus, as a field is not this paradox.