  Derivations are linear operators which satisfy the Leibniz rule, while
integrations are linear operators which satisfy the Rota-Baxter rule. In this
paper, we introduce the notion of an FTC-pair, which consists of an algebra and
module with a derivation and integration between them which together satisfy
analogues of the two Fundamental Theorems of Calculus. In the special case of
when an algebra is seen as a module over itself, we show that this sort of
FTC-pair is precisely the same thing as an integro-differential algebra. We
provide various constructions of FTC-pairs, as well as numerous examples of
FTC-pairs including some based on polynomials, smooth functions, Hurwitz
series, and shuffle algebras. Moreover, it is well known that integrations are
closely related to Zinbiel algebras. The main result of this paper is that the
category of FTC-pairs is equivalent to the category of Zinbiel algebras.
