# Riemann Integral

## What is 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 deficiencies in Riemann integration can be remedied with the Riemann–Stieltjes integral, and most disappear with the Lebesgue integral, though the latter does not have a satisfactory treatment of improper integrals. The gauge integral is a generalisation of the Lebesgue integral that is at once closer to the Riemann integral. These more general theories allow for the integration of more "jagged" or "highly oscillating" functions whose Riemann integral does not exist; but the theories give the same value as the Riemann integral when it does exist.

### Technology Types

bernhard riemanncalculationdefinitiondefinitions of mathematical integrationexplanationintegralproblem solvingprocesspropositionstatementtheoremtheorems in analysistheorythinking

### Synonyms

Lebesgue criterionLebesgue integrability conditionLebesgue integrability criterionLebesgue's criterionLebesgue's integrability criterionRiemann integrabilityRiemann integrableRiemann integrationRiemann-integrabilityRiemann-integrableRiemann's integralRiemannian integral

### Translations

Całka Riemanna (pl)Integración de Riemann (es)Integral de Riemann (ca)Integral de Riemann (pt)Integral Riemann (in)Intégrale de Riemann (fr)Integrale di Riemann (it)Riemannen integral (eu)Riemannintegral (sv)Riemannintegratie (nl)Riemannsches Integral (de)Riemannův integrál (cs)Rimana integralo (eo)Интеграл Римана (ru)Інтеграл Рімана (uk)تكامل ريمان (ar)리만 적분 (ko)リーマン積分 (ja)黎曼积分 (zh)

## Tech Info

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