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The formula expressing the value of a definite integral of a given function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066550/n0665501.png" /> over an interval as the difference of the values at the end points of the interval of any primitive (cf. [[Integral calculus|Integral calculus]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066550/n0665502.png" /> of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066550/n0665503.png" />:
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{{MSC|26A06|26A46}}
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066550/n0665504.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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The formula expressing the value of a definite integral of a given integrable function $f$ over an interval as the difference of the values at the endpoints of the interval of any primitive (cf. [[Integral calculus|Integral calculus]]) $F$ of the function $f$:
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\begin{equation}\label{eq:*}
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\int\limits_a^bf(x)\,dx = F(b)-F(a).
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\end{equation}
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It is named after I. Newton and G. Leibniz, who both knew the rule expressed by \ref{eq:*}, although it was published later. It is also known as "Fundamental theorem of calculus".
  
It is named after I. Newton and G. Leibniz, who both knew the rule expressed by (*), although it was published later.
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If $f$ is [[ Lebesgue integral | Lebesgue integrable]] over $[a,b]$ and $F$ is defined by
 
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\begin{equation*}
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066550/n0665505.png" /> is Lebesgue integrable over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066550/n0665506.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066550/n0665507.png" /> is defined by
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F(x) = \int\limits_a^xf(t)\,dt + C,
 
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\end{equation*}
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066550/n0665508.png" /></td> </tr></table>
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where $C$ is a constant, then $F$ is [[Absolute_continuity#Absolute_continuity_of_a_function | absolutely continuous]], $F'(x) = f(x)$ almost-everywhere on $[a,b]$ (everywhere if $f$ is continuous on $[a,b]$) and \ref{eq:*} is valid.
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066550/n0665509.png" /> is a constant, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066550/n06655010.png" /> is absolutely continuous, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066550/n06655011.png" /> almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066550/n06655012.png" /> (everywhere if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066550/n06655013.png" /> is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066550/n06655014.png" />) and (*) is valid.
 
  
 
A generalization of the Newton–Leibniz formula is the [[Stokes formula|Stokes formula]] for orientable manifolds with a boundary.
 
A generalization of the Newton–Leibniz formula is the [[Stokes formula|Stokes formula]] for orientable manifolds with a boundary.
 
 
 
====Comments====
 
The theorem expressed by the Newton–Leibniz formula is called the fundamental theorem of calculus, cf. e.g. [[#References|[a1]]].
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> K.R. Stromberg,  "Introduction to classical real analysis" , Wadsworth  (1981) pp. 318ff</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Rudin,  "Real and complex analysis" , McGraw-Hill  (1966)  pp. 165ff</TD></TR></table>
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{|
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|-
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|valign="top"|{{Ref|Ru}}|| W. Rudin,  "Real and complex analysis" , McGraw-Hill  (1966).
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|-
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|valign="top"|{{Ref|St}}|| K.R. Stromberg,  "Introduction to classical real analysis" , Wadsworth  (1981).
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|-
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|}

Latest revision as of 20:49, 8 December 2013

2010 Mathematics Subject Classification: Primary: 26A06 Secondary: 26A46 [MSN][ZBL]

The formula expressing the value of a definite integral of a given integrable function $f$ over an interval as the difference of the values at the endpoints of the interval of any primitive (cf. Integral calculus) $F$ of the function $f$: \begin{equation}\label{eq:*} \int\limits_a^bf(x)\,dx = F(b)-F(a). \end{equation} It is named after I. Newton and G. Leibniz, who both knew the rule expressed by \ref{eq:*}, although it was published later. It is also known as "Fundamental theorem of calculus".

If $f$ is Lebesgue integrable over $[a,b]$ and $F$ is defined by \begin{equation*} F(x) = \int\limits_a^xf(t)\,dt + C, \end{equation*} where $C$ is a constant, then $F$ is absolutely continuous, $F'(x) = f(x)$ almost-everywhere on $[a,b]$ (everywhere if $f$ is continuous on $[a,b]$) and \ref{eq:*} is valid.

A generalization of the Newton–Leibniz formula is the Stokes formula for orientable manifolds with a boundary.

References

[Ru] W. Rudin, "Real and complex analysis" , McGraw-Hill (1966).
[St] K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981).
How to Cite This Entry:
Newton-Leibniz formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Newton-Leibniz_formula&oldid=17006
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article