The act of integrating a function and that of differentiating one are inverse operations; thus the area under a curve is an antiderivative of its integrand as a function of the upper end of the area. We explore proof of this claim and its implications for one dimensional integrals.
21.1 The Fundamental Theorem for Ordinary Integrals
21.2 The Fundamental Theorem for Integration in on a Path in the Complex Plane
21.3 The Fundamental Theorem in Integration on a Path in Euclidean Space