Being new to calculus, I'm trying to understand Part 1 of the Fundamental Theorem of Calculus.
Ordinarily, this first part is stated using an " area function" F mapping every x in the domain of f to the number " integral from a to x of f(t)dt".
However, I encounter difficulties to understand what is the status of this area function, being apparently neither an indefinite integral , nor a definite integral( for, I think, a definite integral is a number, not a function); if this " area function" is not an " integral " ( of some sort), I do not understand in which way asserting that F'=f amounts to saying " integration and differentiation are inverse processes" as it is said informally.
Hence my question : is there an easier to understand version of FTC Part 1 that does not make use of the area function concept?
Note : I think I understand in which way the area function is a function and what it " does". What I do not understand is the role it plays in proving that " integration and differentiation a reverse processes" ( being given this function is neither a definite integral, nor an indefinite integral, as MSE answers I got previously tend to show).