Analysis from a categorical perspective
I hesitate to let this out, but there's always this cute little note that I learned from another MO answer (I don't know which one): https://www.maths.ed.ac.uk/~tl/glasgowpssl/banach.pdf. Maybe this will satisfy your curiosity, but I maintain that it takes a warped mind to identify such a categorical formulation of integration as the "right" way to think about integrals.
The advantage of categorical thinking in my view is that it helps to organize computations and arguments involving several different kinds of structures at the same time. For instance, (co)homology is all about capturing useful invariants associated to a complicated structure (e.g. a geometric object) in a much simpler structure (e.g. an abelian group). When we want to determine how the invariants behave under certain operations on the complicated structure (e.g. products, (co)limits) it helps to have a theory already set up to tell us what will happen to the simpler structure. That's where category theory comes into its own, and instances of this paradigm are so ubiquitous in algebra and topology that category theory has taken on a life of its own. It seems that people working in those areas have found it convenient to build categorical constructions into the foundations of their work in order to emphasize generality (one can treat algebraic varieties and solutions to diophantine equations on virtually the same footing), keep track of different notions of equivalence (e.g. homotopy versus homeomorphism), build new kinds of spaces (e.g. groupoids), and to achieve many other aims.
In many kinds of analysis, this kind of abstraction isn't necessary because there's often only one structure to keep track of: $\mathbb{R}$. When you think about it, analysis is only possible because we are willing to seriously overburden $\mathbb{R}$. Take, for example, the expression "$\frac{d}{dt}\int_X f_t(x) d\mu(x)$" and consider all of the different ways real numbers are being used. It is used as a geometric object (odds are X is built out of some construction involving the real numbers or a subspace thereof), a way to give $X$ additional structure (it wouldn't hurt to guess that $\mu$ is a real valued measure), a parameter ($t$), and a reference system ($f$ probably takes values in $\mathbb{R}$ or something related to it). In algebraic geometry, one would probably take each of these roles seriously and understand what kind of structure they are meant to bring to the problem. But part of the power and flexibility of analysis is that we can sweep these considerations under the rug and ultimately reduce most complications to considerations involving the real numbers.
All that being said, the tools of category theory and homological algebra actually have started to make their way into analysis. Because of the fact that analysts generally consider problems tied to certain very specific kinds of structure, they have historically focused on providing the sharpest and most detailed solutions to their problems rather than extracting the crude, qualitative invariants for which cohomological thinking is most appropriate. However, as analysts have become more and more attuned to the deep relationships between functional analysis and geometry, they have turned to ideas from category theory to help keep things organized. K-theory and K-homology have become indispensable tools in operator theory; there is even a bivariant functor $KK(-,-) $ from the category of C-algebras to the category of abelian groups relating the two constructions, and many deep theorems can be subsumed in the assertion that there is a category whose objects are C-algebras and whose morphism spaces are given by $KK(A,B)$. Cyclic homology and cohomology has also become extremely relevant to the interface between analysis and topology.
So ultimately I think it all comes down to what kinds of subtleties are most relevant in a given problem. There is just something fundamentally different about the kind of thinking required to estimate the propagation speed of the solution operator for a nonlinear PDE compared to the kind of thinking required to relate the fixed point theory in characteristic 0 of a linear group acting on a variety to that in characteristic p.
Others can definitely give better opinions, but I currently have "Lectures and exercises on functional analysis" checked out from the library, and I have been enjoying the few parts that I've read so far.
I can not comment on the use of category theory in analysis, but for people who aren't very comfortable with more abstract fields where category theory plays a major role a book like the one above is great since it goes over a lot of basic category theory while keeping the main characters from analysis. At the very least it's a great way to get accustomed to the language.
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
http://math.uga.edu/~pete/243integrals1.pdf
(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.