Probably this is already known to many readers here, but I'll add it because we are in CW mode:
It is possible to construct and characterize $L^1[0,1]$ categorically, namely as the "smallest" pointed Banach space $X$ satisfying the fixed point $X \cong X \times X$. See this note by Tom Leinster and this MO discussion. The integral $\int_{0}^{1} : L^1[0,1] \to \mathbb{R}$ comes from the universal property.
Although I have do admit that this does not compute $\int_{0}^{1} \sin(\pi x) ~ dx$ or alike, I think this is a quite enlightening and strinkingly simple characterization of such a complicated (well at least it is not just a toy example) Banach space. All you have to know is that you can identify a function with two functions, the rest follows. So this example goes in the direction of Paul Garret's answer. I don't know if it simplifies computations or even enables us to do new ones, but it follows one of the goals of category theory: Unification.
