fundamental theorem of calculus

Let $f:[a,b]\to 𝐑$ be a continuous function, let $c\in [a,b]$ be given and consider the integral function $F$ defined on $[a,b]$ as

$$F(x)={\int }_c^{x}f(t)𝑑t.$$

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

$$F^{\prime }(x)=f(x)\mathit{\hspace{1em}\hspace{1em}}\forall x\in [a,b].$$

The previous relation rewritten as

$$\frac{d}{dx}{\int }_c^{x}f(t)𝑑t=f(x)$$

shows that the differentiation operator $\frac{d}{dx}$ is the inverse of the integration operator ${\int }_c^x$. This formula is sometimes called Newton-Leibniz formula.

On the other hand if $f:[a,b]\to 𝐑$ is a continuous function and $G:[a,b]\to 𝐑$ is any antiderivative of $f$, i.e. $G^{\prime }(x)=f(x)$ for all $x\in [a,b]$, then

$${\int }_a^{b}f(t)𝑑t=G(b)-G(a).$$ (1)

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

$${\int }_a^{x}DG=G-G(a).$$

Notes

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

Title	fundamental theorem of calculus
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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