Riemann sums
E47353
Riemann sums are a fundamental method in calculus for approximating the area under a curve by summing the areas of a sequence of rectangles, forming the basis of the definition of the definite integral.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Riemann sums canonical | 2 |
| Darboux sum | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T373790 — 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 sums Context triple: [Bernhard Riemann, knownFor, Riemann sums]
-
A.
Minkowski sum
The Minkowski sum is a fundamental operation in geometry and convex analysis that combines two sets by adding every vector in one set to every vector in the other, widely used in areas such as optimization, robotics, and computational geometry.
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B.
Euler–Maruyama method
The Euler–Maruyama method is a basic time-stepping scheme for numerically approximating solutions to stochastic differential equations, widely used in simulations of systems with noise such as Langevin dynamics.
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C.
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.
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D.
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.
-
E.
Berry–Esseen theorem
The Berry–Esseen theorem is a quantitative refinement of the central limit theorem that provides explicit bounds on the rate of convergence of normalized sums of independent random variables to the normal distribution.
- 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 sums Target entity description: Riemann sums are a fundamental method in calculus for approximating the area under a curve by summing the areas of a sequence of rectangles, forming the basis of the definition of the definite integral.
-
A.
Minkowski sum
The Minkowski sum is a fundamental operation in geometry and convex analysis that combines two sets by adding every vector in one set to every vector in the other, widely used in areas such as optimization, robotics, and computational geometry.
-
B.
Euler–Maruyama method
The Euler–Maruyama method is a basic time-stepping scheme for numerically approximating solutions to stochastic differential equations, widely used in simulations of systems with noise such as Langevin dynamics.
-
C.
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.
-
D.
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.
-
E.
Berry–Esseen theorem
The Berry–Esseen theorem is a quantitative refinement of the central limit theorem that provides explicit bounds on the rate of convergence of normalized sums of independent random variables to the normal distribution.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
construction of the definite integral
ⓘ
mathematical concept ⓘ numerical approximation method ⓘ |
| appearsIn |
introductory calculus courses
ⓘ
real analysis textbooks ⓘ |
| assumes | boundedness of the function on the interval ⓘ |
| basedOn |
partition of an interval
ⓘ
sum of areas of rectangles ⓘ |
| conditionForConvergence | function being Riemann integrable ⓘ |
| contrastWith |
Monte Carlo integration
ⓘ
Simpson's rule ⓘ |
| convergesTo | value of the Riemann integral when the function is Riemann integrable ⓘ |
| coreIdea | approximate area by rectangles over subintervals ⓘ |
| dependsOn |
choice of partition
ⓘ
choice of sample points in each subinterval ⓘ |
| domain | real-valued functions on closed intervals ⓘ |
| field |
calculus
ⓘ
numerical analysis ⓘ real analysis ⓘ |
| generalization |
Riemann–Stieltjes sums
ⓘ
multiple Riemann sums for multivariable integration ⓘ |
| hasComponent |
function values at sample points
ⓘ
partition points ⓘ sample points ⓘ subinterval widths ⓘ |
| hasNotation | sum from i equals 1 to n of f(x_i^*) Δx_i ⓘ |
| hasType |
Riemann sums
self-linksurface differs
ⓘ
surface form:
Darboux sum
left Riemann sum ⓘ lower Riemann sum ⓘ midpoint Riemann sum ⓘ right Riemann sum ⓘ trapezoidal sum ⓘ upper Riemann sum ⓘ |
| introducedIn | 19th century ⓘ |
| limitDefinition | definite integral as limit of Riemann sums ⓘ |
| namedAfter | Bernhard Riemann ⓘ |
| prerequisiteFor | understanding Riemann integration theory ⓘ |
| refinementProperty | finer partitions generally give better approximations ⓘ |
| relatedTo |
Riemann integral
ⓘ
surface form:
Darboux integral
Lebesgue integration ⓘ
surface form:
Lebesgue integral
Riemann integral ⓘ |
| sufficientConditionForIntegrability |
function being bounded and having only finitely many discontinuities
ⓘ
function being continuous on a closed interval ⓘ |
| usedFor |
approximating definite integrals
ⓘ
approximating the area under a curve ⓘ defining the definite integral ⓘ |
| usedIn |
error estimation for integrals
ⓘ
numerical integration ⓘ rigorous proofs of the Fundamental Theorem of Calculus ⓘ |
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 sums Description of subject: Riemann sums are a fundamental method in calculus for approximating the area under a curve by summing the areas of a sequence of rectangles, forming the basis of the definition of the definite integral.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Darboux sum
subject surface form:
Friedrich Bernhard Riemann