{"id":9485,"date":"2024-01-27T16:51:30","date_gmt":"2024-01-27T15:51:30","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=9485"},"modified":"2024-01-27T16:51:40","modified_gmt":"2024-01-27T15:51:40","slug":"construire-un-solide-particulier","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2024\/01\/27\/construire-un-solide-particulier\/","title":{"rendered":"Construire un solide particulier"},"content":{"rendered":"\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">J&rsquo;avais envie de construire un solide particulier en m&rsquo;appuyant sur une courbe aux caract\u00e9ristiques bien pr\u00e9cises&#8230; Voici le r\u00e9sultat:<\/p>\n\n\n\n<figure class=\"wp-block-video\"><video height=\"950\" style=\"aspect-ratio: 1348 \/ 950;\" width=\"1348\" controls src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/01\/solide-2.mp4\"><\/video><\/figure>\n\n\n\n<!--more-->\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_85 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\/2024\/01\/27\/construire-un-solide-particulier\/#Construire_un_solide_particulier_les_contraintes\" >Construire un solide particulier: les contraintes<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/01\/27\/construire-un-solide-particulier\/#Et_si_la_fonction_g_etait_un_polynome\" >Et si la fonction g \u00e9tait un polyn\u00f4me ?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/01\/27\/construire-un-solide-particulier\/#A_la_recherche_de_la_fonction_g\" >\u00c0 la recherche de la fonction g<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/01\/27\/construire-un-solide-particulier\/#Construire_un_solide_particulier_le_volume\" >Construire un solide particulier: le volume<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/01\/27\/construire-un-solide-particulier\/#Quid_de_la_surface_du_solide\" >Quid de la surface du solide?<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Construire_un_solide_particulier_les_contraintes\"><\/span>Construire un solide particulier: les contraintes<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Je voulais avant tout construire une courbe dans un rep\u00e8re du plan. La fonction \\(f\\) devait, dont la courbe devait \u00eatre repr\u00e9sentative, devait imp\u00e9rativement satisfaire les quatre conditions suivantes:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\\( f(x) = g(x)\\text{e}^{-x} \\)<\/li>\n\n\n\n<li>\\( f(0) = 0 \\)<\/li>\n\n\n\n<li>\\( f(7) = 0 \\)<\/li>\n\n\n\n<li>\\( f'(0) = +\\infty \\)<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Et_si_la_fonction_g_etait_un_polynome\"><\/span>Et si la fonction g \u00e9tait un polyn\u00f4me ?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Au d\u00e9but, je cherche \u00e0 faire simple et regarde si un polyn\u00f4me conviendrait pour la fonction <em>g<\/em>&#8230; Mais tr\u00e8s vite, je m&rsquo;aper\u00e7ois que ce n&rsquo;est pas possible \u00e0 cause de la contrainte sur la d\u00e9riv\u00e9e en 0.<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">En effet, la d\u00e9riv\u00e9e d&rsquo;un polyn\u00f4me est toujours un polyn\u00f4me et donc son nombre d\u00e9riv\u00e9 en 0 ne peut pas \u00eatre infini.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"A_la_recherche_de_la_fonction_g\"><\/span>\u00c0 la recherche de la fonction g<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Suite \u00e0 ce constat, je m&rsquo;oriente vers une fonction qui pourrait avant tout satisfaire cette contrainte sur le nombre d\u00e9riv\u00e9 en 0. Il y a le choix entre une fonction inverse et un radical.<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Le deuxi\u00e8me choix semble plus appropri\u00e9 quand on regarde la contrainte de l&rsquo;image de 0 par la fonction <em>f<\/em>&#8230;<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Je vais donc chercher une fonction <em>g<\/em> sous la forme \\( g(x) = \\sqrt{ax^2+bx} \\) (pour satisfaire la contrainte \\(f(0)=0\\). Tr\u00e8s rapidement, \u00e0 l&rsquo;aide de la contrainte \\( f(7) = 0\\), j&rsquo;obtiens:$$ f(x) = \\sqrt{x(7-x)}\\text{e}^{-x}.$$<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">La courbe repr\u00e9sentative de <em>f<\/em> est alors la suivante:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-medium\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/01\/image-1.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"73\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/01\/image-1-300x73.png\" alt=\"construire solide particulier\" class=\"wp-image-9487\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/01\/image-1-300x73.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/01\/image-1-600x147.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/01\/image-1-1024x251.png 1024w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/01\/image-1-768x188.png 768w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/01\/image-1.png 1343w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/figure>\n<\/div>\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Construire_un_solide_particulier_le_volume\"><\/span>Construire un solide particulier: le volume<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Si on engendre un solide \u00e0 partir de cette courbe en la faisant tourner autour de l&rsquo;axe des abscisses, on obtient ceci:<\/p>\n\n\n\n<figure class=\"wp-block-video\"><video height=\"950\" style=\"aspect-ratio: 1348 \/ 950;\" width=\"1348\" autoplay loop src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/01\/solide-2.mp4\"><\/video><\/figure>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Je souhaite \u00e0 pr\u00e9sente calculer le volume de ce solide. Pour cela, je consid\u00e8re que le volume est la somme infinit\u00e9simale d&rsquo;aires de disques de rayon \\(f(x)\\), pour \\(x \\in [0;7]\\).<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-medium\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/01\/image-3.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"98\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/01\/image-3-300x98.png\" alt=\"\" class=\"wp-image-9490\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/01\/image-3-300x98.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/01\/image-3-600x195.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/01\/image-3-1024x333.png 1024w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/01\/image-3-768x250.png 768w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/01\/image-3.png 1514w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/figure>\n<\/div>\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Le volume est donc:$$\\mathcal{V} = \\int_0^7 \\pi \\times \\big[ f(x) \\big]^2 \\text{d}x = \\pi\\int_0^7 x(7-x)\\text{e}^{-x}\\text{d}x.$$; <\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Bon, l\u00e0, un petit coup d&rsquo;int\u00e9gration par parties devrait faire l&rsquo;affaire! J&rsquo;explique ? Allez, j&rsquo;explique. Je pose:$$\\begin{array}{ll} u'(x)=\\text{e}^{-x}\\qquad &amp; v(x)=-x^2+7x \\\\ u(x)=-\\text{e}^{-x} &amp; v'(x)=-2x+7 \\end{array}$$On a alors:<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">$$\\begin{align}\\mathcal{v} &amp; = \\pi\\int_0^7 u'(x)v(x) \\text{d}x\\\\ &amp; = \\pi \\big[ u(x)v(x) \\big]_0^7 &#8211; \\pi\\int_0^7 u(x)v'(x)\\text{d}x\\\\ &amp; = \\pi\\big( (uv)(7) &#8211; (uv)(0) \\big) &#8211;  \\pi\\int_0^7 u(x)v'(x)\\text{d}x\\\\ &amp; = &#8211; \\pi\\int_0^7 u(x)v'(x)\\text{d}x \\\\ &amp; = \\pi\\int_0^7 (-2x+7)\\text{e}^{-x}. \\end{align}$$<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Une deuxi\u00e8me int\u00e9gration par parties s&rsquo;impose en posant:$$\\begin{array}{ll} u'(x)=\\text{e}^{-x}\\qquad &amp; v(x)=-2x+7 \\\\ u(x)=-\\text{e}^{-x} &amp; v'(x)=-2 \\end{array}$$<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">On obtient:$$\\begin{align} \\mathcal{V} &amp; = \\pi \\big[ (uv)(7)-(uv)(0) \\big] &#8211; \\pi\\int_0^7 u(x)v'(x)\\text{d}x\\\\ &amp; = \\pi( 7\\text{e}^{-7} + 7) +2 \\pi\\int_0^7 \\text{e}^{-x}\\text{d}x\\\\ &amp; = \\pi( 7\\text{e}^{-7} + 7) -2 \\pi\\big[ -\\text{e}^{-x} \\big]_0^7 \\\\ &amp; = \\pi( 7\\text{e}^{-7} + 7) -2 \\pi\\big( -\\text{e}^{-7} + 1 \\big)\\\\&amp;=\\big(9\\text{e}^{-7}+5\\big)\\pi.\\end{align}$$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Quid_de_la_surface_du_solide\"><\/span>Quid de la surface du solide?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">La surface du solide est obtenue en multipliant la longueur de la courbe sur [0;7] par \\(2\\pi\\). La longueur d&rsquo;une courbe est donn\u00e9e par la formule:$$\\mathcal{L}=\\int_0^7 \\sqrt{1+f(x)^2}\\text{d}x = \\int_0^7 \\sqrt{1+x(7-x)\\text{e}^{-2x}} \\text{d}x.$$<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Et l\u00e0, c&rsquo;est le drame&#8230; ! Fort heureusement, nous avons de bons outils pour en donner une approximation (nous ne sommes plus au Moyen-\u00c2ge!). J&rsquo;utilise personnellement le logiciel <a href=\"https:\/\/www-fourier.ujf-grenoble.fr\/~parisse\/install_fr\" target=\"_blank\" rel=\"noreferrer noopener\">Xcas<\/a>.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-medium\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/01\/image-4.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"86\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/01\/image-4-300x86.png\" alt=\"\" class=\"wp-image-9501\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/01\/image-4-300x86.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/01\/image-4-600x172.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/01\/image-4-768x220.png 768w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/01\/image-4.png 902w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/figure>\n<\/div>\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Oui, parce que bon&#8230; Le calcul th\u00e9orique de l&rsquo;int\u00e9grale me semble bien compliqu\u00e9&#8230;<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">En multipliant cette valeur par \\(2\\pi\\), on arrive \u00e0 une surface d&rsquo;environ 48 unit\u00e9s d&rsquo;aires.<\/p>\n\n\n\n<p class=\"is-style-Paragraph-paragraph wp-block-paragraph\">Ainsi, si on d\u00e9cide de construire un tel objet, en prenant comme unit\u00e9 de longueur le centim\u00e8tre, et si l&rsquo;on souhaite le peindre, on saura qu&rsquo;il faut environ 48 cm\u00b2 de peinture.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>J&rsquo;avais envie de construire un solide particulier en m&rsquo;appuyant sur une courbe aux caract\u00e9ristiques bien pr\u00e9cises&#8230; Voici le r\u00e9sultat:<\/p>\n","protected":false},"author":1,"featured_media":9502,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[104,409,105,143],"class_list":["post-9485","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematiques","tag-integrales","tag-integration-par-parties","tag-volume","tag-xcas"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.9 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Construire un solide particulier - Mathweb.fr<\/title>\n<meta name=\"description\" content=\"J&#039;avais envie de construire un solide particulier en m&#039;appuyant sur une courbe aux caract\u00e9ristiques bien pr\u00e9cises. 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