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The Fundamental Theorem of Calculus (Evaluation Theorem)

In this video we prove the FTC (which we refer to as the "Evaluation Theorem") – namely, if F'(x) = f(x), then the definite integral of f on [a,b] is equal to F(b) - F(a). The proof depends heavily on the notion of "decorated partitions", Riemann sums, and the "existence theorem" for definite integrals, topics covered previously in the video "Riemann sums and the definite integral (    • Riemann sums and the definite integral   ). We test drive the theorem on a handful of examples and discuss the consistency of the theorem with "null integration" and the "limit swap law" for limits of integration.