Fundamental Theorem of Calculus

E259760

mathematical theorem theorem of calculus

The Fundamental Theorem of Calculus links differentiation and integration by showing that the definite integral of a function can be computed using any of its antiderivatives.

Try in SPARQL Jump to: Surface forms Disambiguation Statements Elicitation Referenced by

All labels observed (7)

Label	Occurrences
First Fundamental Theorem of Calculus	2
Second Fundamental Theorem of Calculus	2
Fundamental Theorem of Calculus canonical	1
Fundamental Theorem of Calculus, Part 1	1
Fundamental Theorem of Calculus, Part 2	1
Fundamental theorem of calculus	1
fundamental theorem of calculus	1

How this entity was disambiguated

This entity first appeared as the object of triple T2364426 — 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: Fundamental Theorem of Calculus
Context triple: [Riemann integral, relatedTo, Fundamental Theorem of Calculus]

A. Riemann integral
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.
B. Riemann sums
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.
C. Cauchy integral theorem
The Cauchy integral theorem is a fundamental result in complex analysis stating that the integral of a holomorphic function over any closed contour in a simply connected domain is zero.
D. Itô’s lemma
Itô’s lemma is a fundamental result in stochastic calculus that generalizes the chain rule to functions of stochastic processes, especially Brownian motion.
E. Riemann–Liouville integral
The Riemann–Liouville integral is a fundamental operator in fractional calculus that generalizes the concept of an n-fold repeated integral to non-integer (fractional) orders.
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: Fundamental Theorem of Calculus
Target entity description: The Fundamental Theorem of Calculus links differentiation and integration by showing that the definite integral of a function can be computed using any of its antiderivatives.

A. Riemann integral
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.
B. Riemann sums
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.
C. Cauchy integral theorem
The Cauchy integral theorem is a fundamental result in complex analysis stating that the integral of a holomorphic function over any closed contour in a simply connected domain is zero.
D. Itô’s lemma
Itô’s lemma is a fundamental result in stochastic calculus that generalizes the chain rule to functions of stochastic processes, especially Brownian motion.
E. Riemann–Liouville integral
The Riemann–Liouville integral is a fundamental operator in fractional calculus that generalizes the concept of an n-fold repeated integral to non-integer (fractional) orders.
F. None of above. chosen

Statements (46)

Predicate	Object
instanceOf	mathematical theorem ⓘ mathematical theorem ⓘ mathematical theorem ⓘ theorem of calculus ⓘ
alsoKnownAs	FTC ⓘ
appliesTo	Riemann integrable functions ⓘ continuous real-valued functions on closed intervals ⓘ
assumesCondition	integrand is typically continuous on a closed interval ⓘ
category	real analysis theorem ⓘ
connects	Riemann integral and derivative ⓘ
developedInCentury	17th century ⓘ
field	calculus ⓘ mathematical analysis ⓘ
formalStatement	If f is continuous on [a,b] and F is an antiderivative of f on [a,b], then ∫_a^b f(x) dx = F(b) − F(a) ⓘ If f is continuous on [a,b] and F is any antiderivative of f on [a,b], then ∫_a^b f(x) dx = F(b) − F(a) ⓘ If f is integrable on [a,b] and F(x)=∫_a^x f(t) dt, then F is continuous on [a,b], differentiable on (a,b), and F′(x)=f(x) for all x in (a,b) ⓘ
generalizedBy	Henstock–Kurzweil integral versions ⓘ Lebesgue version of the fundamental theorem of calculus ⓘ Stieltjes integral versions ⓘ
hasConsequence	fundamental relationship between area under a curve and accumulation of rates of change ⓘ linearity of the definite integral is compatible with antiderivatives ⓘ
hasPart	First Fundamental Theorem of Calculus ⓘ linked to: Fundamental Theorem of Calculus Second Fundamental Theorem of Calculus ⓘ linked to: Fundamental Theorem of Calculus
historicallyAttributedTo	Gottfried Wilhelm Leibniz ⓘ Isaac Newton ⓘ
implies	definite integral can be evaluated using antiderivatives ⓘ existence of antiderivative for continuous functions defined by an integral ⓘ
isTaughtIn	advanced placement calculus curricula ⓘ introductory calculus courses ⓘ university analysis courses ⓘ
relates	indefinite integral and definite integral ⓘ
relatesConcept	differentiation ⓘ integration ⓘ
requires	notion of definite integral ⓘ notion of derivative ⓘ
shows	evaluation of definite integrals reduces to evaluating antiderivatives at endpoints ⓘ every continuous function on a closed interval has an antiderivative defined by an integral ⓘ
statesRoughly	differentiation and definite integration are inverse processes ⓘ
underlies	many solution methods for differential equations ⓘ standard techniques of integration ⓘ
usedFor	accumulation function analysis ⓘ area computation ⓘ computing definite integrals ⓘ solving problems in engineering ⓘ solving problems in physics ⓘ solving problems in probability theory ⓘ

How these facts were elicited