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In the introduction to Chap. 10 the notion of the definite integral of a function f on an interval [a, b] was already mentioned. It arises from summing up expressions of the form \(f(x)\varDelta x\) 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.