{"id":11631,"date":"2026-04-20T17:06:00","date_gmt":"2026-04-20T15:06:00","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=11631"},"modified":"2026-04-20T17:08:16","modified_gmt":"2026-04-20T15:08:16","slug":"methode-de-simpson-integration-et-paraboles","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2026\/04\/20\/methode-de-simpson-integration-et-paraboles\/","title":{"rendered":"M\u00e9thode de Simpson: int\u00e9gration et paraboles"},"content":{"rendered":"\n<p>La m\u00e9thode de Simpson est une m\u00e9thode d&#8217;int\u00e9gration qui permet d&#8217;obtenir une approcimation d&#8217;une int\u00e9grale \u00e0 l&#8217;aide de paraboles.<\/p>\n\n\n\n<!--more-->\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 counter-hierarchy ez-toc-counter ez-toc-white ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Au menu sur cette page...<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.mathweb.fr\/euclide\/2026\/04\/20\/methode-de-simpson-integration-et-paraboles\/#Methode_dintegration_de_Simpson_introduction\" >M\u00e9thode d&#8217;int\u00e9gration de Simpson: introduction<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.mathweb.fr\/euclide\/2026\/04\/20\/methode-de-simpson-integration-et-paraboles\/#Methode_de_Simpson_la_formule\" >M\u00e9thode de Simpson: la formule<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.mathweb.fr\/euclide\/2026\/04\/20\/methode-de-simpson-integration-et-paraboles\/#Polynome_dinterpolation_de_Lagrange\" >Polyn\u00f4me d&#8217;interpolation de Lagrange<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.mathweb.fr\/euclide\/2026\/04\/20\/methode-de-simpson-integration-et-paraboles\/#La_formule_composite_de_Simpson\" >La formule (composite) de Simpson<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.mathweb.fr\/euclide\/2026\/04\/20\/methode-de-simpson-integration-et-paraboles\/#Implementation_en_Python\" >Impl\u00e9mentation en Python<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Methode_dintegration_de_Simpson_introduction\"><\/span>M\u00e9thode d&#8217;int\u00e9gration de Simpson: introduction<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>J&#8217;ai d\u00e9j\u00e0 r\u00e9dig\u00e9 quelques articles sur d&#8217;autres m\u00e9thodes d&#8217;approximation d&#8217;une int\u00e9grale (m\u00e9thode des rectangles, des trap\u00e8zes, du point m\u00e9dian), qui font appel \u00e0 des polygones. <\/p>\n\n\n\n<p>La m\u00e9thode de Simpson va plus loin: elle approxime localement une courbe par un morceau de parabole.<\/p>\n\n\n\n<p>Prenons une fonction $f$ d\u00e9finie sur un intervalle $[a;b]$. On subdivise cet intervalle en intervalles $[x_0=a;x_2]$, $[x_2;x_4]$, &#8230;, $[x_{2n-2};x_{2n}=b]$. Sur chaque intervalle $[x_{2k};x_{2(k+1)}]$, on choisit alors trois points qui ont pour coordonn\u00e9es:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>$(x_{2k};f(x_{2k}))$<\/li>\n\n\n\n<li>$(x_{2k+1};f(x_{2k+1}))$, avec $m=\\frac{x_{2k}+x_{2(k+1)}}{2}$<\/li>\n\n\n\n<li>$(x_{2(k+1)};f(x_{2(k+1)}))$<\/li>\n<\/ul>\n\n\n\n<p>L&#8217;id\u00e9e consiste alors \u00e0 construire la parabole passant par ces trois points. L&#8217;aire sous cette parabole servira d&#8217;approximation de $\\displaystyle\\int_{x_{2k}}^{x_{2(k+1)}} f(x)\\text{d}x$.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure data-wp-context=\"{&quot;imageId&quot;:&quot;69e68350e3be5&quot;}\" data-wp-interactive=\"core\/image\" data-wp-key=\"69e68350e3be5\" class=\"aligncenter size-medium wp-lightbox-container\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"222\" data-wp-class--hide=\"state.isContentHidden\" data-wp-class--show=\"state.isContentVisible\" data-wp-init=\"callbacks.setButtonStyles\" data-wp-on--click=\"actions.showLightbox\" data-wp-on--load=\"callbacks.setButtonStyles\" data-wp-on-window--resize=\"callbacks.setButtonStyles\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2026\/04\/image-300x222.png\" alt=\"m\u00e9thode Simpson composite\" class=\"wp-image-11635\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2026\/04\/image-300x222.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2026\/04\/image-1024x757.png 1024w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2026\/04\/image-768x568.png 768w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2026\/04\/image.png 1173w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><button\n\t\t\tclass=\"lightbox-trigger\"\n\t\t\ttype=\"button\"\n\t\t\taria-haspopup=\"dialog\"\n\t\t\taria-label=\"Agrandir\"\n\t\t\tdata-wp-init=\"callbacks.initTriggerButton\"\n\t\t\tdata-wp-on--click=\"actions.showLightbox\"\n\t\t\tdata-wp-style--right=\"state.imageButtonRight\"\n\t\t\tdata-wp-style--top=\"state.imageButtonTop\"\n\t\t>\n\t\t\t<svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"12\" height=\"12\" fill=\"none\" viewBox=\"0 0 12 12\">\n\t\t\t\t<path fill=\"#fff\" d=\"M2 0a2 2 0 0 0-2 2v2h1.5V2a.5.5 0 0 1 .5-.5h2V0H2Zm2 10.5H2a.5.5 0 0 1-.5-.5V8H0v2a2 2 0 0 0 2 2h2v-1.5ZM8 12v-1.5h2a.5.5 0 0 0 .5-.5V8H12v2a2 2 0 0 1-2 2H8Zm2-12a2 2 0 0 1 2 2v2h-1.5V2a.5.5 0 0 0-.5-.5H8V0h2Z\" \/>\n\t\t\t<\/svg>\n\t\t<\/button><\/figure>\n<\/div>\n\n\n<p>Comme on peut le voir sur le sch\u00e9ma, l&#8217;intervalle $[a;b]$ est subdivis\u00e9 en un nombre <em>pair<\/em> d&#8217;intervalles.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Methode_de_Simpson_la_formule\"><\/span>M\u00e9thode de Simpson: la formule<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Polynome_dinterpolation_de_Lagrange\"><\/span>Polyn\u00f4me d&#8217;interpolation de Lagrange<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>Avant de regarder ce que donne la formule engendr\u00e9e par la m\u00e9thode de Simpson, il faut parler de ce polyn\u00f4me. <\/p>\n\n\n\n<p>Consid\u00e9rons $n+1$ points de coordonn\u00e9es $(x_k;f(x_k))$, $0 \\leq k \\leq n$. Si on souhaite construire un polyn\u00f4me dont la courbe repr\u00e9sentative passe par tous ces points, on utilise la formule de Lagrange:$$L(x)=\\sum_{i=0}^n f(x_i)\\prod_{\\substack{k=0\\\\k\\neq i}}^n \\frac{x-x_k}{x_k-x_i}.$$<\/p>\n\n\n\n<p>Appliqu\u00e9 \u00e0 3 points $x_k$, $x_{k+1}$ et $x_{k+2}$, cela donne:$$\\begin{align*}P_k(x)&amp;=\\frac{x-x_{k+1}}{x_k-x_{k+1}}\\times \\frac{x-x_{k+2}}{x_k-x_{k+2}}f(x_{k})\\\\&amp; + \\frac{x-x_{k}}{x_{k+1}-x_{k}}\\times \\frac{x-x_{k+2}}{x_{k+1}-x_{k+2}}f(x_{k+1}) \\\\ &amp; + \\frac{x-x_{k}}{x_{k+2}-x_{k}}\\times \\frac{x-x_{k+1}}{x_{k+2}-x_{k+1}}f(x_{k+2}).\\end{align*}$$<\/p>\n\n\n\n<p>N&#8217;oublions pas que $x_{k+1}-x_k=x_{k+2}-x_{k+1}=h$, ce qui simplifie l&#8217;\u00e9criture du polyn\u00f4me de Lagrange:$$\\begin{align*}P_k(x)&amp;=\\frac{(x-x_{k}-h)(x-x_k-2h)}{2h^2}f(x_k) + \\frac{(x-x_{k})(x-x_k-2h)}{h^2}f(x_{k+1})\\\\&amp; + \\frac{(x-x_{k})(x-x_k-h)}{2h^2}f(x_{k+2}).\\end{align*}$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"La_formule_composite_de_Simpson\"><\/span>La formule (composite) de Simpson<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>On va poser $t=x-x_k$. Alors, le polyn\u00f4me devient:$$P_k(t)=\\frac{(t-h)(t-2h)}{2h^2}f(x_k)-\\frac{t(t-2h)}{h^2}f(x_{k+1})+\\frac{t(t-h)}{2h^2}f(x_{k+2}).$$ De plus,<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>quand $x=x_k$, $t=0$;<\/li>\n\n\n\n<li>quand $x=x_{k+2}$, $t=x_{k+2}-x_k=2h$;<\/li>\n\n\n\n<li>$\\text{d}t = \\text{d}x$.<\/li>\n<\/ul>\n\n\n\n<p>Ainsi,$$\\begin{align*}\\int_{x_{k}}^{x_{k+2}}P_k(x)\\text{d}x &amp; = \\frac{f(x_k)}{2h^2}\\int_0^{2h} (t-h)(t-2h)  ~\\text{d}t &#8211; \\frac{f(x_{k+1})}{h^2}\\int_0^{2h} t(t-2h)~\\text{d}t \\\\&amp; + \\frac{f(x_{k+2})}{2h^2}\\int_0^{2h} t(t-h)~\\text{d}t\\\\\\int_{x_{k}}^{x_{k+2}}P_k(x)\\text{d}x&amp;= \\frac{f(x_k)}{2h^2}\\int_0^{2h} (t^2-3ht+2h^2)  ~\\text{d}t\\\\ &amp; &#8211; \\frac{f(x_{k+1})}{h^2}\\int_0^{2h} (t^2-2ht)~\\text{d}t \\\\&amp; + \\frac{f(x_{k+2})}{2h^2}\\int_0^{2h} (t^2-ht)~\\text{d}t\\\\ \\int_{x_{k}}^{x_{k+2}}P_k(x)\\text{d}x&amp;=\\frac{f(x_k)}{2h^2}\\left[ \\frac{t^3}{3}-\\frac{3ht^2}{2}+2h^2t\\right]_0^{2h} \\\\ &amp; &#8211; \\frac{f(x_{k+1})}{h^2}\\left[ \\frac{t^3}{3}-ht^2\\right]_0^{2h} \\\\ &amp; + \\frac{f(x_{k+2})}{2h^2}\\left[ \\frac{t^3}{3} &#8211; \\frac{ht^2}{2} \\right]_0^{2h}\\\\ \\int_{x_{k}}^{x_{k+2}}P_k(x)\\text{d}x&amp;= \\frac{h}{3}f(x_k) + \\frac{4h}{3}f(x_{k+1}) + \\frac{h}{3}f(x_{k+2})\\\\ &amp; = \\frac{h}{3}\\big( f(x_k) + 4f(x_{k+1}) + f(x_{k+2})\\big)\\end{align*}$$<\/p>\n\n\n\n<p>\u00c0 ce stade, on n&#8217;oublie pas que $h=\\frac{b-a}{2n}$.<\/p>\n\n\n\n<p>On arrive alors \u00e0 l&#8217;approximation:$$\\boxed{\\int_a^b f(x)~\\text{d}x \\approx \\frac{b-a}{6n}\\sum_{k=0}^{n-1}\\big( f(x_k) + 4f(x_{k+1}) + f(x_{k+2})\\big)}$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Implementation_en_Python\"><\/span>Impl\u00e9mentation en Python<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">from math import log, exp\n\ndef f(x):\n    return 5 * log(x) * exp(-x\/2)\n\ndef simpson(a,b,n):\n    S = 0\n    h = (b-a)\/(2*n)\n    for k in range(n):\n        x0 = a + 2*k*h\n        x1 = x0 + h\n        x2 = x0 + 2*h\n        S += f(x0) + 4*f(x1) + f(x2)\n        \n    return h * S \/ 3<\/pre>\n\n\n\n<pre class=\"wp-block-code\"><code>&gt;&gt;&gt; print( simpson(1,10,10) )\n5.4286825969998596<\/code><\/pre>\n\n\n<div class=\"wp-block-image\">\n<figure data-wp-context=\"{&quot;imageId&quot;:&quot;69e68350e45e6&quot;}\" data-wp-interactive=\"core\/image\" data-wp-key=\"69e68350e45e6\" class=\"aligncenter size-medium wp-lightbox-container\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"76\" data-wp-class--hide=\"state.isContentHidden\" data-wp-class--show=\"state.isContentVisible\" data-wp-init=\"callbacks.setButtonStyles\" data-wp-on--click=\"actions.showLightbox\" data-wp-on--load=\"callbacks.setButtonStyles\" data-wp-on-window--resize=\"callbacks.setButtonStyles\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2026\/04\/image-1-300x76.png\" alt=\"\" class=\"wp-image-11654\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2026\/04\/image-1-300x76.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2026\/04\/image-1-768x194.png 768w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2026\/04\/image-1.png 959w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><button\n\t\t\tclass=\"lightbox-trigger\"\n\t\t\ttype=\"button\"\n\t\t\taria-haspopup=\"dialog\"\n\t\t\taria-label=\"Agrandir\"\n\t\t\tdata-wp-init=\"callbacks.initTriggerButton\"\n\t\t\tdata-wp-on--click=\"actions.showLightbox\"\n\t\t\tdata-wp-style--right=\"state.imageButtonRight\"\n\t\t\tdata-wp-style--top=\"state.imageButtonTop\"\n\t\t>\n\t\t\t<svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"12\" height=\"12\" fill=\"none\" viewBox=\"0 0 12 12\">\n\t\t\t\t<path fill=\"#fff\" d=\"M2 0a2 2 0 0 0-2 2v2h1.5V2a.5.5 0 0 1 .5-.5h2V0H2Zm2 10.5H2a.5.5 0 0 1-.5-.5V8H0v2a2 2 0 0 0 2 2h2v-1.5ZM8 12v-1.5h2a.5.5 0 0 0 .5-.5V8H12v2a2 2 0 0 1-2 2H8Zm2-12a2 2 0 0 1 2 2v2h-1.5V2a.5.5 0 0 0-.5-.5H8V0h2Z\" \/>\n\t\t\t<\/svg>\n\t\t<\/button><figcaption class=\"wp-element-caption\">Avec Geogebra<\/figcaption><\/figure>\n<\/div>\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>La m\u00e9thode de Simpson est une m\u00e9thode d&#8217;int\u00e9gration qui permet d&#8217;obtenir une approcimation d&#8217;une int\u00e9grale \u00e0 l&#8217;aide de paraboles.<\/p>\n","protected":false},"author":1,"featured_media":11655,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[],"class_list":["post-11631","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-python"],"yoast_head":"<!-- This site is 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