In rigorously defining the antiderivative as an operator from a function to a class of functions (hence the infamous +C), and then using a representative from the ensuing antiderivative aka integral to calculate area, the question arises as to which representative from the class should be used to calculate area. This is clear how it works in practice, for individual functions. But in a general definition, I am not so sure. I am looking for some clearer theoretical characterization than the typical "apply these rules and then take C=0". For example, if we were only dealing with polynomials, I could say that one takes the representative that passes through the origin. But this does not work for, say, all trigonometric functions.