Here are some unpolemical facts concerning the Riemann integral: 1) The Riemann integral has a geometric interpretation which is different than that of the Lebesgue integral and is certainly useful in some places. For a bounded set $S \subset \mathbb{R}^N$, Riemann integrability of the characte...

This community wiki answer is addressed to the OP's comment that he is looking for an "axiomatic" approach to the integral. I don't (yet) understand what axioms have to do with category theory. In particular, with respect to the example you give, I don't see what is particularly categorical abo...

I wish to respectfully point out a suboptimality in your approach: you are trying to prove the Fundamental Theorem of Calculus by yourself and by talking to a (possibly not very knowledgeable, but I don't know the whole story) teacher. This is not necessary: there are literally thousands of book...

It's natural that the Fundamental Theorem of Calculus has two parts, since morally it expresses the fact that differentiation and integration are mutually inverse processes, and this amounts to two statements: (i) integrating and then differentiating and (ii) differentiating and then integrating ...

In his notes: http://math.uga.edu/~pete/243integrals1.pdf, Pete Clark outlines an axiomatic approach to the Riemann Integral. He doesn't use the language of sheafs, but it seems implicit in his definition before Theorem 1. He goes on to show that the fundamental theorem of calculus follows, and ...