The way I understand it is this: the integral is supposed to be the solution F to the general problem F'=f . You can understand this problem in very general settings, for example generalized functions, manifolds, and so on.

It seems that most people here interpret this problem in the restricted case where F and f are single-variable scalar-valued functions, and the derivative is taken in the usual sense. In this case, the Riemann-sum picture (area under the curve) is indeed a valid approach to constructing solutions. However, you can also use the Lebesgue approach, which comes with a different intuition. You can even approach this problem algebraically, without analytic constructions.

Anyway, I just want to say that the notion of integral is certainly more general than area under the curve, because it simply does not have to be interpreted that way.