We can think of the pullback of fiber bundle an integral as follows. Suppose $p:E \to B$ is a fiber bundle and $f:A \to B$ is a map. Then we have the pullback bundle $A \times_f E \to A$. The bundle over $A$ is in a sense approximated by $\bigsqcup_i A_i \times p^{-1}(f(a_i))$ where the $A_i$ partition $A$ into small sets and $a_i \in A$. By taking the limit of these "Riemann sum bundles" we get $A \times_f E$. The notation "$\times_f$" is consistent with the interpretation of the integral as generalized multiplication. It is interesting to note that in category theory ends generalize limits which generalize pullbacks (and products) and the ends are denoted in part by an integral sign, but this is probably irrelevant to my question.
My question is if and how this interpretation of the pullback can be useful for anything beyond itself (and I suppose the notation for ends). For instance, you integrate differential forms and you can pullback characteristic classes, but is there any deeper connection than the fact the terms "integrate" and "pullback" show up in both?