Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus establishes that differentiation and integration are inverse operations — that the rate of change of an accumulated quantity is the quantity itself, and that the accumulation of a rate of change recovers the original function up to a constant. In formal terms, if F is an antiderivative of f, then the definite integral of f from a to b equals F(b) − F(a). This is not merely a computational convenience; it is the statement that local behavior (the derivative) and global behavior (the integral) are structurally unified.

The theorem has two parts, usually attributed to Newton and Leibniz independently: the first part states that differentiation reverses integration; the second part provides the computational method for evaluating definite integrals via antiderivatives. Before the theorem, integration was understood as summation — the method of exhaustion, of Riemann sums, of laborious geometric calculation. After the theorem, integration became the inverse of differentiation, and the two operations were recognized as dual aspects of a single structure.

The Fundamental Theorem is often taught as a technical result, but its deeper significance is epistemological: it guarantees that the study of change and the study of accumulation are the same study. Any discipline that tracks how things evolve and how things add up is implicitly relying on this theorem — whether its practitioners know it or not.