Riemann integral

E47347

integral mathematical concept notion in real analysis

The Riemann integral is a fundamental concept in calculus that defines the integral of a function as the limit of sums of function values over increasingly fine partitions of an interval.

Jump to: Surface forms Statements Referenced by

Observed surface forms (2)

Surface form	Occurrences
Darboux integral	1
Riemann integration	1

Statements (51)

Predicate	Object
instanceOf	integral ⓘ mathematical concept ⓘ notion in real analysis ⓘ
approximatedBy	Riemann sums ⓘ Simpson's rule ⓘ midpoint rule ⓘ trapezoidal rule ⓘ
basedOn	limit of Riemann sums ⓘ
characterizedBy	Cauchy criterion for Riemann integrability ⓘ equality of upper and lower Darboux integrals ⓘ
closedUnder	addition of integrable functions ⓘ scalar multiplication of integrable functions ⓘ
codomain	real numbers ⓘ
contrastedWith	Henstock–Kurzweil integral ⓘ Lebesgue integral ⓘ
defines	integral of a real-valued function on an interval ⓘ
domain	functions defined on closed bounded intervals of real numbers ⓘ
equivalentTo	Darboux integral for bounded functions on closed intervals ⓘ
failsToIntegrate	some bounded functions with dense discontinuities ⓘ
field	calculus ⓘ real analysis ⓘ
generalizationOf	area under a curve ⓘ finite sums ⓘ
hasDefinition	limit of sums of function values times subinterval lengths as mesh size tends to zero ⓘ
hasVariant	Riemann–Stieltjes integral ⓘ improper Riemann integral ⓘ
implies	function is Riemann integrable on the interval ⓘ
integrates	all continuous functions on closed bounded intervals ⓘ bounded functions with sets of discontinuities of measure zero ⓘ piecewise continuous functions on closed bounded intervals ⓘ
introducedBy	Bernhard Riemann ⓘ
namedAfter	Bernhard Riemann ⓘ
notation	∫_a^b f(x) dx ⓘ
relatedTo	Fundamental Theorem of Calculus ⓘ
requires	boundedness of the function on the interval ⓘ existence of a limit of Riemann sums independent of choice of tags ⓘ
satisfies	absolute value inequality ⓘ additivity over intervals ⓘ linearity ⓘ monotonicity ⓘ
subsetOf	Lebesgue integrable functions on a finite interval ⓘ
taughtIn	undergraduate calculus courses ⓘ
usedFor	computing accumulated quantities ⓘ computing areas under curves ⓘ defining average value of a function on an interval ⓘ
uses	lower sums ⓘ partitions of an interval ⓘ tagged partitions ⓘ upper sums ⓘ
weakerThan	Lebesgue integral in terms of generality ⓘ
yearIntroduced	1854 ⓘ

Referenced by (7)

Full triples — surface form annotated when it differs from this entity's canonical label.

Lebesgue integration → generalizes → Riemann integral ⓘ

this entity surface form: Riemann integration

Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe → introducedConcept → Riemann integral ⓘ

Bernhard Riemann → knownFor → Riemann integral ⓘ

Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe → mainTopic → Riemann integral ⓘ

Friedrich → notableWork → Riemann integral ⓘ

subject surface form: Friedrich Bernhard Riemann

Georg → notableWork → Riemann integral ⓘ

subject surface form: Georg Friedrich Bernhard Riemann

Riemann sums → relatedTo → Riemann integral ⓘ

this entity surface form: Darboux integral