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 about the Eilenberg-Steenrod axioms (unless you mean to count the functorial nature of co/homology as one of the axioms).
As an example of an axiomatic treatment of the (Riemann) integral, see Section 2 of
(Note: this is nothing very original. For instance, shortly after I wrote this I saw that Lang had almost the same treatment in his undergraduate analysis text.)
Here I see no category theory whatsoever. Is this what you had in mind? Why or why not?
Perhaps you were talking about the Lebesgue integral rather than the Riemann integral. In that respect, I would say that the Daniell approach to the Lebesgue integral (i.e., characterizing it in terms of the completion of a certain normed linear space) feels "axiomatic" to me but still not categorical.