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fundamental theorem of calculus


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

F(x)=∫xcf(t)𝑑t.

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

Fβ€²(x)=f(x)  

The previous relationMathworldPlanetmath rewritten as

dd⁒x⁒∫cxf⁒(t)⁒𝑑t=f⁒(x)

shows that the differentiationMathworldPlanetmath operator dd⁒x 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 function and 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:

∫axD⁒G=G-G⁒(a).

Notes

Equation (1) is sometimes called β€œBarrow’s rule” or β€œBarrow’s formula”.

Title fundamental theorem of calculusMathworldPlanetmathPlanetmath
Canonical name FundamentalTheoremOfCalculus
Date of creation 2013-03-22 14:13:27
Last modified on 2013-03-22 14:13:27
Owner paolini (1187)
Last modified by paolini (1187)
Numerical id 13
Author paolini (1187)
Entry type Theorem
Classification msc 26A42
Synonym Newton-Leibniz
Synonym Barrow’s rule
Synonym Barrow’s formula
Related topic FundamentalTheoremOfCalculus
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