This article provides another point of view on the theory of rough paths, which starts with simple considerations on ordinary integrals, and endows theimportance of the Green-Riemann formula, as in the work of D. Feyel and A. de La Pradelle. This point of view allows us to introduce gently the required algebraic structures and provides alternative ways to understand why the construction provided by T. Lyons et al. is a natural generalization of the notion of integral of differential forms, in the sense it shares the same properties as integrals along smooth paths, when we use the ``right notion'' of path.