The Fundamental Theorem of Calculus

Abstract

In Chap. 9 we evaluated a few definite integrals by using Riemann sums, a method that is generally not easy, even for simple integrands. Then we evaluated a few more, by interpreting the definite integral as a certain familiar area. Generally however, computing definite integrals which routinely appear in problems with no apparent notion of area or of average value in sight, can be very difficult. Coming to the rescue in many cases is the Fundamental Theorem of Calculus. With it, many more definite integrals can be computed relatively easily. But this—the most important theorem in all of Calculus—gives us a great deal more.