Abstract

In this thesis, we describe the historical development of the modern integration theory by presenting the key ideas and insights that led to its shaping, from the first rigorous definition of the definite integral given by Cauchy in 1823, to Lebesgue's theory as it first appeared in his doctoral thesis of 1902. In the final part we also present recent approaches in integration theory. We show that various problems motivated the enrichment of the notion of integral, while one in particular constituted the most important trigger of this development from the beginnings well into the 20th century: the search for a better understanding of Fourier series. For our study, we principally look at original sources, and we provide detailed proofs for important results, often by elaborating on the authors' sketchy or heuristic arguments, malting thus the original results accessible to the modern reader. We then use this historical and epistemological analysis to raise some issues related to the teaching of integration at various levels at university, i.e., in calculus, analysis and measure theory courses. For this, we look at some typical textbooks and, inspired by the historical analysis, we give some suggestions for teaching. Our findings show that there might be some conceptual gaps between subsequent levels that are not necessarily insuperable, but require careful didactical analysis.