In the first semester of calculus two types of integrals are taught to students: definite integral and indefinite integral (a.k.a. antiderivative). They are proven to be related through the fundamental theorem of calculus.
Later, both these notions are generalized. Antiderivation can be placed into context of differential forms: $\psi \in \Omega^n(M)$ is called an antiderivative of $\omega \in \Omega^{n+1}(M)$, if $d\psi = \omega$. Definite integral, on the other hand, can be viewed as pairing operation for dual objects: functions and measures or, even more generally, differential forms and currents (I should probably have specified some assumptions of regularity here, but I'd rather avoid it for now). These two notions of integral are still related via the general Stokes' theorem.
There is, however, a third face for the concept — I am talking about an integral of a differential equation. That is, a function that stays constant along the trajectories. In physics it is interpreted as a conserved quantity (e.g., energy, momentum, etc.). In one-dimensional setting it is very directly related to the first two notions, but in higher dimensions this connection becomes less clear. Nevertheless, I want to believe, that there exists a united point of view, encompassing all three ideas.
Do you know anything like that?
