Riemann integral [SEP] named after
Riemann integral
In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Göttingen in 1854, but not published in a journal until 1868. For many functions and practical applications, the Riemann integral can be evaluated by the fundamental theorem of calculus or approximated by numerical integration.
The Riemann integral is unsuitable for many theoretical purposes. Some of the technical
introductory textbooks on calculus and real analysis often develop Riemann integration using the Darboux integral, rather than the true Riemann integral. Moreover, the definition is readily extended to defining Riemann-Stieltjes integration. Darboux integrals are named after their inventor, Gaston Darboux.
Definition.
The definition of the Darboux integral considers upper and lower (Darboux) integrals, which exist for any bounded real-valued function "f" on the interval ["a","b"]. The Darboux integral exists if and only if the upper and lower