proof of the fundamental theorem of calculus

Recall that a continuous functionMathworldPlanetmathPlanetmath is Riemann integrablePlanetmathPlanetmath on every interval [c,x], so the integral


is well defined.

Consider the increment of F:


(we have used the linearity of the integral with respect to the function and the additivity with respect to the domain).

Since f is continuous, by the mean-value theorem, there exists ξh∈[x,x+h] such that f⁒(ξh)=F⁒(x+h)-F⁒(x)h so that


since ξh→x as h→0. This proves the first part of the theorem.

For the second part suppose that G is any antiderivative of f, i.e.Β Gβ€²=f. Let F be the integral function


We have just proven that Fβ€²=f. So F′⁒(x)=G′⁒(x) for all x∈[a,b] or, which is the same, (G-F)β€²=0. This means that G-F is constant on [a,b] that is, there exists k such that G⁒(x)=F⁒(x)+k. Since F⁒(a)=0 we have G⁒(a)=k and hence G⁒(x)=F⁒(x)+G⁒(a) for all x∈[a,b]. Thus

Title proof of the fundamental theorem of calculusMathworldPlanetmathPlanetmath
Canonical name ProofOfTheFundamentalTheoremOfCalculus
Date of creation 2013-03-22 13:45:37
Last modified on 2013-03-22 13:45:37
Owner paolini (1187)
Last modified by paolini (1187)
Numerical id 10
Author paolini (1187)
Entry type Proof
Classification msc 26-00