For a function f (x) continuous over the interval [ a, b ], with F (x) as its antiderivative, the integral of f (x) over [ a, b ] is equal to F (b) minus F (a): . In effect, the first fundamental theorem of calculus defines the existence of an antiderivative for a continuous function.efinition of First Fundamental Theorem of Calculus. The first fundamental theorem of calculus describes the relationship between differentiation and integration, which are inverse functions of one another.