I've gone all the way up to calculus 3 (vectors, multi-dimensional functions, and doing differentials and integrals therein) and I've yet to see calculus applied by any programming. I am curious how one actually implements it though, in what (limited) programming I've done, I haven't seen any clear way to calculate say an integral using something like c++ or c#.

Doing real math on a computer is actually surprisingly complicated. The topic is usually referred to as Numerical Analysis, and if you're looking to take a class in it you'll find it in the upper division course work of the Applied Math degree track, and you'll almost certainly be expected to have had at least an introductory course in Differential Equations (essentially, Calc 4) first.

You may also find an introduction to the topic in the lower-division math requirements of most CS degrees, in classes typically titled as Discrete or Finite Math.

That said, yes there are libraries, although the folks who are serious about that stuff are more likely to use dedicated packages/environments/languages like MatLab, Mathematica, or R.

I will also say that I think there's a level of dishonesty when people say they aren't using math as programmers or software engineers. Sure, I'm not solving quadratics or writing proofs, but I am certainly using the concepts that I learned on a daily basis. Graph theory, group theory, set theory, number theory, linear algebra... all of these things matter when you're trying to create software that actually does something useful. Just because it doesn't look the same as it did on the final exam of that class doesn't mean it isn't the same kind of math.