In the introduction to Chap. 10 the notion of the definite integral of a function f on an interval [a,b] has already been mentioned. It arises from summing up expressions of the form f(x)Δx and taking limits. Such sums appear in many applications including the calculation of areas, surface areas and volumes as well as the calculation of lengths of curves. This chapter employs the notion of Riemann integrals as the basic concept of definite integration. Riemann’s approach provides an intuitive concept in many applications, as will be elaborated in examples at the end of the chapter.
The main part of Chap. 11 is dedicated to the properties of the integral. In particular, the two fundamental theorems of calculus are proven. The first theorem allows one to calculate a definite integral from the knowledge of an antiderivative. The second fundamental theorem states that the definite integral of a function f on an interval [a,x] with variable upper bound provides an antiderivative of f. Since the definite integral can be approximated, for example by Riemann sums, the second fundamental theorem offers a possibility to approximate the antiderivative numerically. This is of importance, for example, for the calculation of distribution functions in statistics.