Since F(x) is an anti-derivative of f(x) which can vary by an added constant, F(a) can be interpreted as the area from some arbitrary constant c to a where c varies by choice of additive constant, NOT f(x).
For example, if then an anti-derivative, F(x), is of the form where C can be any constant. If, say, C= 0 then is the area under the curve from 0 to a. If we were to take C= 9, then is the area under the curve from -3 to a.
(Remember that "area under the curve" is one possible interpretation of the anti-derivative. It isn't really correct to say that an anti-derivative is an area.)