<div class="wims_chemin">\reload{<img src="gifs/doc/etoile.gif" alt="rechargez" width="20" height="20" border=0>}\link{main}{Intgration numrique} <img src="gifs/arrows/right3.32.gif" alt=" ---> " width="25" height="15" border=0 valign="bottom"> \link{mainS1}{I  Introduction} <img src="gifs/arrows/right3.32.gif" alt=" ---> " width="25" height="15" border=0 valign="bottom"> I-3  Rsultats fondamentaux</div><table width=100%><tr><td valign=top><div class="left_toc"><p>
<div class="left_selection">\link{mainS1}{I  Introduction}</div>

\link{mainS2}{II  Formules de quadrature et leur ordre}

\link{mainS3}{III  Mise en oeuvre sur Matlab}

\link{mainS4}{IV  Etude de l'erreur d'une mthode de quadrature}

\link{mainS5}{V  Exemples de calcul numrique de l'ordre}

\link{mainS6}{VI  Bibliographie}

\link{mainS7}{VII  Exercices}


\link{index}{Index}</div></td><td valign=top align=left width=100%><div class="wimsdoc">
<h2 class="thm">Proposition</h2><div class="thm">
Si \(  f: \left [a ,\;  b \right ] \longrightarrow \mathbb R \) est Riemann intgrable, alors
<div class="math">\(\displaystyle  \int^{b}_{a}f(x)\;dx = \displaystyle \lim_{|\sigma| \longrightarrow 0}
S_{\sigma} (f) = \displaystyle \lim_{|\sigma| \longrightarrow 0}
s_{\sigma} (f)\)</div> ou d'une manire quivalente 
<div class="math">\(
\forall \varepsilon > 0, \; \exists \eta > 0 \; \mbox { tq
} \; \forall \sigma, \;
| \sigma | < \eta \Longrightarrow \begin{matrix}  
|I(f) - S_{\sigma}(f)| & \leq &  \varepsilon \\
|I(f) - s_{\sigma}(f)| & \leq &  \varepsilon 
\end{matrix} 
\)</div>
</div>



<h2 class="rmq">Remarque</h2><div class="rmq">
Si \(  f:
\left [a ,\;  b \right ] \longrightarrow \mathbb R\;  \) est continue alors <div class="math">\(\; \displaystyle \inf_{x \in
  \left ]x_i ,\;  x_{i+1} \right [ } f(x) =  \displaystyle \inf_{x \in
  \left [x_i ,\;  x_{i+1} \right ] } f(x)\; \mbox { et } \; \displaystyle \sup_{x \in
  \left ]x_i ,\;  x_{i+1} \right [ } f(x) =  \displaystyle \sup_{x \in
  \left [x_i ,\;  x_{i+1} \right ] } f(x)\)</div> 
</div>



<h2 class="thm">Thorme</h2><div class="thm">
Si \(  f:
\left [a ,\;  b \right ] \longrightarrow \mathbb R\; \)  est continue alors <div class="math">\(\; \displaystyle
\int^{b}_{a}f(x)\;dx = \displaystyle \lim_{n \longrightarrow +\infty} \displaystyle \frac{b-a}{n}
\displaystyle \sum_{i=0}^{n-1}m_i = \displaystyle \lim_{n \longrightarrow +\infty} \displaystyle \frac{b-a}{n}
\displaystyle \sum_{i=0}^{n-1}M_i\)</div> et d'une faon plus gnrale  
<div class="math">\(
\int^{b}_{a}f(x)\;dx = \displaystyle \lim_{n \longrightarrow +\infty} \displaystyle \frac{b-a}{n}
\displaystyle \sum_{i=0}^{n-1} f(c_i) \; \mbox{ avec } \; c_i \in [x_i, \; x_{i+1}]
\)</div>
</div>

</div></td><td valign=top align=right> <div class="right_toc">
\link{mainS1S1}{I-1  Problme tudi}

\link{mainS1S2}{I-2  Notations et dfinitions}

<div class="right_selection">\link{mainS1S3}{I-3  Rsultats fondamentaux}</div>
</div><center>\reload{<img src="gifs/doc/etoile.gif" alt="rechargez" width="20" height="20" border=0>}</center></td></tr></table>