Riemann integral
E47347
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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Riemann integral canonical | 9 |
| Riemann integration | 2 |
| Darboux integral | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T373778 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Riemann integral Context triple: [Bernhard Riemann, knownFor, Riemann integral]
-
A.
Gaussian integral
The Gaussian integral is a fundamental result in mathematics that evaluates the integral of the exponential of a negative quadratic function over the entire real line, yielding a value proportional to the square root of π and underpinning the normal distribution in probability theory.
-
B.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
-
C.
inverse function theorem
The inverse function theorem is a fundamental result in calculus and differential geometry that gives conditions under which a differentiable function has a locally defined differentiable inverse near a point where its derivative is invertible.
-
D.
Itô calculus
Itô calculus is a branch of stochastic analysis that extends classical calculus to functions of stochastic processes, particularly Brownian motion, enabling rigorous treatment of stochastic differential equations.
-
E.
implicit function theorem
The implicit function theorem is a fundamental result in calculus and differential geometry that guarantees, under suitable smoothness and nondegeneracy conditions, the local solvability of equations for some variables as differentiable functions of others.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Riemann integral Target entity description: 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.
-
A.
Gaussian integral
The Gaussian integral is a fundamental result in mathematics that evaluates the integral of the exponential of a negative quadratic function over the entire real line, yielding a value proportional to the square root of π and underpinning the normal distribution in probability theory.
-
B.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
-
C.
inverse function theorem
The inverse function theorem is a fundamental result in calculus and differential geometry that gives conditions under which a differentiable function has a locally defined differentiable inverse near a point where its derivative is invertible.
-
D.
Itô calculus
Itô calculus is a branch of stochastic analysis that extends classical calculus to functions of stochastic processes, particularly Brownian motion, enabling rigorous treatment of stochastic differential equations.
-
E.
implicit function theorem
The implicit function theorem is a fundamental result in calculus and differential geometry that guarantees, under suitable smoothness and nondegeneracy conditions, the local solvability of equations for some variables as differentiable functions of others.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Riemann integral Description of subject: 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.
Referenced by (12)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Darboux integral
Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe
→
mainTopic
→
Riemann integral
ⓘ
Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe
→
introducedConcept
→
Riemann integral
ⓘ
subject surface form:
Georg Friedrich Bernhard Riemann
this entity surface form:
Riemann integration
subject surface form:
Friedrich Bernhard Riemann
this entity surface form:
Riemann integration