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fundamental theorem of calculus (Theorem)

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

$\displaystyle F(x)= \int_c^x f(t)\, dt. $

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

$\displaystyle F'(x)=f(x)\qquad \forall x\in [a,b]. $

The previous relation rewritten as

$\displaystyle \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\colon[a,b]\to \mathbf R$ is a continuous function and $ G\colon[a,b]\to \mathbf R$ is any antiderivative of $ f$, i.e. $ G'(x)=f(x)$ for all $ x\in[a,b]$, then

$\displaystyle \int_a^b f(t) \, dt = G(b)-G(a).$ (1)

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

$\displaystyle \int_a^x D G = G - G(a). $

Notes

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



"fundamental theorem of calculus" is owned by paolini.
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See Also: fundamental theorems of calculus for Lebesgue integration, fundamental theorem of calculus for Kurzweil-Henstock integral, fundamental theorem of calculus for Riemann integration

Other names:  Newton-Leibniz, Barrow's rule, Barrow's formula

Attachments:
proof of the fundamental theorem of calculus (Proof) by paolini
substitution notation (Topic) by pahio
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Cross-references: equation, derivative, inverse, operator, differentiation, relation, differentiable, antiderivative, function, integral, continuous function
There are 11 references to this entry.

This is version 10 of fundamental theorem of calculus, born on 2004-03-01, modified 2008-04-06.
Object id is 5660, canonical name is FundamentalTheoremOfCalculusClassicalVersion.
Accessed 8366 times total.

Classification:
AMS MSC: 26A42 (Real functions :: Functions of one variable :: Integrals of Riemann, Stieltjes and Lebesgue type)
Pending Errata and Addenda
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Why does this entry exist? by archibal on 2004-03-03 06:56:50
We already have an entry on the Fundamental Theorem of Calculus which covers both fundamental theorems of calculus. Its generality is slightly less in some ways (requires absolute continuity) but this could be fixed. What is the need for a second entry?
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