In the US we'd reserve "mathematical analysis" (or more specifically, Real analysis) to the college level classes which involve writing proofs about the continuity of functions between sets of real numbers. You'd probably end up with a lecture on the mean value theorem here, and leave with the ability to prove it, among other things

"Calculus" is the application of that theory without argument. It's an advanced high school class or an early college one. There you'll integrate or differentiate real valued functions for use in optimisation problems or for determining qualitative features of such a function (e.g. where is it flat, where is it defined, etc).

In the US, you can probably pass calculus without writing a proof, but you can't pass mathematical analysis without at least understanding epsilon/delta proofs.