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


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


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 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:



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