The theorem transforms motion into accumulation and accumulation back into motion.

The Fundamental Theorem of Calculus connects the two central operations of calculus: differentiation and integration. It shows that finding the accumulated quantity represented by an integral can be reversed by taking a derivative, and vice versa. In effect, rates of change and total accumulation are two sides of the same mathematical coin. This connection turns many difficult area and accumulation problems into manageable calculations using antiderivatives. It also provides the theoretical foundation for evaluating definite integrals without endlessly summing tiny pieces. Elegant and surprisingly powerful, it unifies calculus into a coherent whole. It is used primarily in physics, engineering, and applied mathematics.