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单词 FundamentalTheoremOfCalculus
释义

fundamental theorem of calculus


Let f:[a,b]𝐑 be a continuous functionMathworldPlanetmathPlanetmath, let c[a,b] be givenand consider the integral function F defined on [a,b] as

F(x)=cxf(t)𝑑t.

Then F is an antiderivative of f that is,F is differentiableMathworldPlanetmathPlanetmath in [a,b] and

F(x)=f(x)  x[a,b].

The previous relationMathworldPlanetmath rewritten as

ddxcxf(t)𝑑t=f(x)

shows that the differentiationMathworldPlanetmath operator ddx is the inversePlanetmathPlanetmathPlanetmath of the integration operator cx. This formulaMathworldPlanetmathPlanetmath is sometimes called Newton-Leibniz formula.

On the other hand if f:[a,b]𝐑 is a continuous functionand G:[a,b]𝐑 is any antiderivative of f, i.e. G(x)=f(x) for all x[a,b], then

abf(t)𝑑t=G(b)-G(a).(1)

This shows that up to a constant, the integration operator is the inverse of the derivativePlanetmathPlanetmath operator:

axDG=G-G(a).

Notes

Equation (1) is sometimes called “Barrow’s rule” or “Barrow’s formula”.

Titlefundamental theorem of calculusMathworldPlanetmathPlanetmath
Canonical nameFundamentalTheoremOfCalculus
Date of creation2013-03-22 14:13:27
Last modified on2013-03-22 14:13:27
Ownerpaolini (1187)
Last modified bypaolini (1187)
Numerical id13
Authorpaolini (1187)
Entry typeTheorem
Classificationmsc 26A42
SynonymNewton-Leibniz
SynonymBarrow’s rule
SynonymBarrow’s formula
Related topicFundamentalTheoremOfCalculus
Related topicFundamentalTheoremOfCalculusForKurzweilHenstockIntegral
Related topicFundamentalTheoremOfCalculusForRiemannIntegration
Related topicLaplaceTransformOfFracftt
Related topicLimitsOfNaturalLogarithm
Related topicFundamentalTheoremOfIntegralCalculus
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更新时间:2025/5/3 12:29:37