A construção da integral de Riemann e o Teorema Fundamental do Cálculo

Abstract

The following monograph aims to conduct a bibliographical study on the construction of the Riemann integral and the Fundamental Theorem of Calculus within the framework of Real Analysis. To establish the concepts, it first presents a historical overview of the problem of determining areas, from the geometric methods of Antiquity to the formulations developed by Riemann and Darboux, and includes brief biographical accounts of mathematicians who contributed to the development of the theory. Subsequently, it systematizes the fundamental concepts of the field of real numbers, the topological properties of the real line, limits, continuity, and differentiability, which are essential for understanding integration. The study initially presents the integral through Darboux’s approach by introducing partitions, lower and upper sums, lower and upper integrals, and the criterion for integrability. Then, it presents Riemann sums, tagged partitions, and the norm of a partition, culminating in the formal definition of the Riemann integral. Finally, it states and proves the Fundamental Theorem of Calculus, in addition to presenting an application.