Journal Article: The Riemann Integral – Foundations, Developments, and Generalizations
Abstract The Riemann integral, introduced in the 19th century by Bernhard Riemann, remains a cornerstone of real analysis and mathematical integration. Defined through the limiting process of Riemann sums over partitions of an interval, it provides a framework for integrating bounded functions on closed intervals. Despite its elegance, the Riemann integral has limitations, particularly with functions exhibiting dense discontinuities. This article explores the classical definition, its properties, limitations, and modern generalizations such as the Lebesgue and Kurzweil–Henstock integrals, situating the Riemann integral within the broader landscape of integration theory. e-Journals math.nie.edu.sg Springer 1. Introduction Integration is a fundamental concept in mathematics, bridging geometry, analysis, and applied sciences. The Riemann integral, formulated in 1854, was the first rigorous definition of integration beyond geometric intuition. It remains widely taught due to its a...
