Longitud de arco - Wikipedia, la enciclopedia libre

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class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Métodos_anteriores_al_cálculo"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Métodos anteriores al cálculo</span> </div> </a> <button aria-controls="toc-Métodos_anteriores_al_cálculo-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Alternar subsección Métodos anteriores al cálculo</span> </button> <ul id="toc-Métodos_anteriores_al_cálculo-sublist" class="vector-toc-list"> <li id="toc-Antigüedad" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Antigüedad"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Antigüedad</span> </div> </a> <ul id="toc-Antigüedad-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Siglo_XVII" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Siglo_XVII"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Siglo <span>XVII</span></span> </div> </a> <ul id="toc-Siglo_XVII-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Véase_también" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Véase_también"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Véase también</span> </div> </a> <ul id="toc-Véase_también-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Enlaces_externos" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Enlaces_externos"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Enlaces externos</span> </div> </a> <ul id="toc-Enlaces_externos-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contenidos" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Cambiar a la tabla de contenidos" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Cambiar a la tabla de contenidos</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Longitud de arco</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Ir a un artículo en otro idioma. Disponible en 35 idiomas" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-35" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">35 idiomas</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8B%A8%E1%8C%8E%E1%89%A3%E1%8C%A3_%E1%88%AD%E1%8B%9D%E1%88%98%E1%89%B5" title="የጎባጣ ርዝመት (amárico)" lang="am" hreflang="am" data-title="የጎባጣ ርዝመት" data-language-autonym="አማርኛ" data-language-local-name="amárico" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B7%D9%88%D9%84_%D9%82%D9%88%D8%B3" title="طول قوس (árabe)" lang="ar" hreflang="ar" data-title="طول قوس" data-language-autonym="العربية" data-language-local-name="árabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C6%8Fyrinin_uzunlu%C4%9Fu" title="Əyrinin uzunluğu (azerbaiyano)" lang="az" hreflang="az" data-title="Əyrinin uzunluğu" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaiyano" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%94%D0%B0%D1%9E%D0%B6%D1%8B%D0%BD%D1%8F_%D0%BA%D1%80%D1%8B%D0%B2%D0%BE%D0%B9" title="Даўжыня крывой (bielorruso)" lang="be" hreflang="be" data-title="Даўжыня крывой" data-language-autonym="Беларуская" data-language-local-name="bielorruso" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%94%D0%B0%D1%9E%D0%B6%D1%8B%D0%BD%D1%8F_%D0%BA%D1%80%D1%8B%D0%B2%D0%BE%D0%B9" title="Даўжыня крывой (Belarusian (Taraškievica orthography))" lang="be-tarask" hreflang="be-tarask" data-title="Даўжыня крывой" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Longitud_d%27arc" title="Longitud d'arc (catalán)" lang="ca" hreflang="ca" data-title="Longitud d'arc" data-language-autonym="Català" data-language-local-name="catalán" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/D%C3%A9lka_k%C5%99ivky" title="Délka křivky (checo)" lang="cs" hreflang="cs" data-title="Délka křivky" data-language-autonym="Čeština" data-language-local-name="checo" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Kurvel%C3%A6ngde" title="Kurvelængde (danés)" lang="da" hreflang="da" data-title="Kurvelængde" data-language-autonym="Dansk" data-language-local-name="danés" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/L%C3%A4nge_(Mathematik)" title="Länge (Mathematik) (alemán)" lang="de" hreflang="de" data-title="Länge (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="alemán" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9C%CE%AE%CE%BA%CE%BF%CF%82_%CF%84%CF%8C%CE%BE%CE%BF%CF%85" title="Μήκος τόξου (griego)" lang="el" hreflang="el" data-title="Μήκος τόξου" data-language-autonym="Ελληνικά" data-language-local-name="griego" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Arc_length" title="Arc length (inglés)" lang="en" hreflang="en" data-title="Arc length" data-language-autonym="English" data-language-local-name="inglés" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Pikkus_(matemaatika)" title="Pikkus (matemaatika) (estonio)" lang="et" hreflang="et" data-title="Pikkus (matemaatika)" data-language-autonym="Eesti" data-language-local-name="estonio" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B7%D9%88%D9%84_%D9%82%D9%88%D8%B3" title="طول قوس (persa)" lang="fa" hreflang="fa" data-title="طول قوس" data-language-autonym="فارسی" data-language-local-name="persa" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/K%C3%A4yr%C3%A4n_pituus" title="Käyrän pituus (finés)" lang="fi" hreflang="fi" data-title="Käyrän pituus" data-language-autonym="Suomi" data-language-local-name="finés" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Longueur_d%27un_arc" title="Longueur d'un arc (francés)" lang="fr" hreflang="fr" data-title="Longueur d'un arc" data-language-autonym="Français" data-language-local-name="francés" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/%C3%8Dvhossz" title="Ívhossz (húngaro)" lang="hu" hreflang="hu" data-title="Ívhossz" data-language-autonym="Magyar" data-language-local-name="húngaro" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Ferillengd" title="Ferillengd (islandés)" lang="is" hreflang="is" data-title="Ferillengd" data-language-autonym="Íslenska" data-language-local-name="islandés" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Lunghezza_di_un_arco" title="Lunghezza di un arco (italiano)" lang="it" hreflang="it" data-title="Lunghezza di un arco" data-language-autonym="Italiano" data-language-local-name="italiano" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%BC%A7%E9%95%B7" title="弧長 (japonés)" lang="ja" hreflang="ja" data-title="弧長" data-language-autonym="日本語" data-language-local-name="japonés" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B3%A1%EC%84%A0%EC%9D%98_%EA%B8%B8%EC%9D%B4" title="곡선의 길이 (coreano)" lang="ko" hreflang="ko" data-title="곡선의 길이" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Booglengte" title="Booglengte (neerlandés)" lang="nl" hreflang="nl" data-title="Booglengte" data-language-autonym="Nederlands" data-language-local-name="neerlandés" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Buelengde" title="Buelengde (noruego bokmal)" lang="nb" hreflang="nb" data-title="Buelengde" data-language-autonym="Norsk bokmål" data-language-local-name="noruego bokmal" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/D%C5%82ugo%C5%9B%C4%87_krzywej" title="Długość krzywej (polaco)" lang="pl" hreflang="pl" data-title="Długość krzywej" data-language-autonym="Polski" data-language-local-name="polaco" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Comprimento_do_arco" title="Comprimento do arco (portugués)" lang="pt" hreflang="pt" data-title="Comprimento do arco" data-language-autonym="Português" data-language-local-name="portugués" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%94%D0%BB%D0%B8%D0%BD%D0%B0_%D0%BA%D1%80%D0%B8%D0%B2%D0%BE%D0%B9" title="Длина кривой (ruso)" lang="ru" hreflang="ru" data-title="Длина кривой" data-language-autonym="Русский" data-language-local-name="ruso" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Duljina_luka" title="Duljina luka (serbocroata)" lang="sh" hreflang="sh" data-title="Duljina luka" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbocroata" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Arc_length" title="Arc length (Simple English)" lang="en-simple" hreflang="en-simple" data-title="Arc length" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Dol%C5%BEina_loka" title="Dolžina loka (esloveno)" lang="sl" hreflang="sl" data-title="Dolžina loka" data-language-autonym="Slovenščina" data-language-local-name="esloveno" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Gjat%C3%ABsia_e_harkut" title="Gjatësia e harkut (albanés)" lang="sq" hreflang="sq" data-title="Gjatësia e harkut" data-language-autonym="Shqip" data-language-local-name="albanés" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%94%D1%83%D0%B6%D0%B8%D0%BD%D0%B0_%D0%BB%D1%83%D0%BA%D0%B0" title="Дужина лука (serbio)" lang="sr" hreflang="sr" data-title="Дужина лука" data-language-autonym="Српски / srpski" data-language-local-name="serbio" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/B%C3%A5gl%C3%A4ngd" title="Båglängd (sueco)" lang="sv" hreflang="sv" data-title="Båglängd" data-language-autonym="Svenska" data-language-local-name="sueco" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B8%A2%E0%B8%B2%E0%B8%A7%E0%B8%AA%E0%B9%88%E0%B8%A7%E0%B8%99%E0%B9%82%E0%B8%84%E0%B9%89%E0%B8%87" title="ความยาวส่วนโค้ง (tailandés)" lang="th" hreflang="th" data-title="ความยาวส่วนโค้ง" data-language-autonym="ไทย" data-language-local-name="tailandés" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D0%BE%D0%B2%D0%B6%D0%B8%D0%BD%D0%B0_%D0%BA%D1%80%D0%B8%D0%B2%D0%BE%D1%97" title="Довжина кривої (ucraniano)" lang="uk" hreflang="uk" data-title="Довжина кривої" data-language-autonym="Українська" data-language-local-name="ucraniano" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%BC%A7%E9%95%BF" title="弧长 (chino wu)" lang="wuu" hreflang="wuu" data-title="弧长" data-language-autonym="吴语" data-language-local-name="chino wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%BC%A7%E9%95%BF" title="弧长 (chino)" lang="zh" hreflang="zh" data-title="弧长" data-language-autonym="中文" data-language-local-name="chino" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span 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data-event-name="pinnable-header.vector-appearance.unpin">ocultar</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">De Wikipedia, la enciclopedia libre</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="es" dir="ltr"><div class="rellink noprint hatnote"> Para otros usos de este término, véase <a href="/wiki/Longitud" title="Longitud">Longitud</a>.</div> <p>En <a href="/wiki/Matem%C3%A1tica" class="mw-redirect" title="Matemática">matemática</a>, la <b>longitud de arco</b>, también llamada <b>rectificación de una curva</b>, es la medida de la <a href="/wiki/Distancia" title="Distancia">distancia</a> o <i>camino recorrido</i> a lo largo de una <a href="/wiki/Curva" title="Curva">curva</a> o dimensión lineal. Históricamente, ha sido difícil determinar esta longitud en <a href="/wiki/Segmento" title="Segmento">segmentos</a> irregulares; aunque fueron utilizados varios métodos para curvas específicas. La llegada del <a href="/wiki/C%C3%A1lculo" title="Cálculo">cálculo</a> trajo consigo la fórmula general para obtener soluciones cerradas para algunos casos. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Cálculo_mediante_integrales"><span id="C.C3.A1lculo_mediante_integrales"></span>Cálculo mediante integrales</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Longitud_de_arco&action=edit&section=1" title="Editar sección: Cálculo mediante integrales"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Al considerar una curva definida por una <a href="/wiki/Funci%C3%B3n_(matem%C3%A1ticas)" class="mw-redirect" title="Función (matemáticas)">función</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> y su respectiva <a href="/wiki/Derivada" title="Derivada">derivada</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0cd7d7c75340e779d82658e19d1720ce84ab127" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.144ex; height:3.009ex;" alt="{\displaystyle f'(x)}"></span> que son continuas en un intervalo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}"></span>, la longitud <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> del arco delimitado por <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> es dada por la ecuación: </p> <blockquote style="padding: 5px 10px; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align: left; margin-left:30px; margin-bottom: 0.4em; margin-top:0.2em; min-width:50%;"> <p><span style="float: right; width: 10%; text-align: right;">(<cite id="Equation_1" style="font-style: normal;"><a href="#Eqnref_1">1</a></cite>)</span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=\int _{a}^{b}{\sqrt {1+\left[f'\left(x\right)\right]^{2}}}\,{\text{d}}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mo>[</mo> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=\int _{a}^{b}{\sqrt {1+\left[f'\left(x\right)\right]^{2}}}\,{\text{d}}x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a72b677a4de88c9b80aa8fa96c3e6da15f240d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.193ex; height:6.343ex;" alt="{\displaystyle s=\int _{a}^{b}{\sqrt {1+\left[f'\left(x\right)\right]^{2}}}\,{\text{d}}x}"></span> </p> </blockquote> <p>En el caso de una curva definida paramétricamente mediante dos funciones dependientes de <i>t</i> como <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=f\left(t\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=f\left(t\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76f96d773f517c7f1910615e77ae170793f0bd15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.743ex; height:2.843ex;" alt="{\displaystyle x=f\left(t\right)}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=g\left(t\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>g</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=g\left(t\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7823b71109799e4cd2cd5254bbafef6005bf9987" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.406ex; height:2.843ex;" alt="{\displaystyle y=g\left(t\right)}"></span>, la longitud del arco desde el punto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f(a),g(a))\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f(a),g(a))\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c3b325b9fac3e226e57b6e5113fe485568a807e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.703ex; height:2.843ex;" alt="{\displaystyle (f(a),g(a))\,}"></span> hasta el punto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f(b),g(b))\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f(b),g(b))\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/927d9ba7480255eb0b74f9174e1570027dcb5e8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.239ex; height:2.843ex;" alt="{\displaystyle (f(b),g(b))\,}"></span> se calcula mediante: </p> <blockquote style="padding: 5px 10px; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align: left; margin-left:30px; margin-bottom: 0.4em; margin-top:0.2em; min-width:50%;"> <p><span style="float: right; width: 10%; text-align: right;">(<cite id="Equation_2" style="font-style: normal;"><a href="#Eqnref_2">2</a></cite>)</span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=\int _{a}^{b}{\sqrt {\left[f'\left(t\right)\right]^{2}+\left[g'\left(t\right)\right]^{2}}}\,{\text{d}}t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow> <mo>[</mo> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>[</mo> <mrow> <msup> <mi>g</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=\int _{a}^{b}{\sqrt {\left[f'\left(t\right)\right]^{2}+\left[g'\left(t\right)\right]^{2}}}\,{\text{d}}t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/696fad607e0d418a2a980a566d3cfc9dc4f27d6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.237ex; height:6.343ex;" alt="{\displaystyle s=\int _{a}^{b}{\sqrt {\left[f'\left(t\right)\right]^{2}+\left[g'\left(t\right)\right]^{2}}}\,{\text{d}}t}"></span> </p> </blockquote> <p>Si la función está definida por coordenadas polares donde la coordenadas radial y el ángulo polar están relacionados mediante <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=f(\theta )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=f(\theta )\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1650cc0e3a6fd23f0f9e7c964c9728c76169e872" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.713ex; height:2.843ex;" alt="{\displaystyle r=f(\theta )\,}"></span>, la longitud del arco comprendido en el intervalo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\alpha ,\beta ]\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">]</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [\alpha ,\beta ]\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f11c36714bc748bec4f07fa119a9de430438d34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.534ex; height:2.843ex;" alt="{\displaystyle [\alpha ,\beta ]\,}"></span>, toma la forma: </p> <blockquote style="padding: 5px 10px; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align: left; margin-left:30px; margin-bottom: 0.4em; margin-top:0.2em; min-width:50%;"> <p><span style="float: right; width: 10%; text-align: right;">(<cite id="Equation_3" style="font-style: normal;"><a href="#Eqnref_3">3</a></cite>)</span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=\int _{\alpha }^{\beta }{\sqrt {[f(\theta )]^{2}+\left[f'(\theta )\right]^{2}}}\,{\text{d}}\theta \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>[</mo> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>θ</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=\int _{\alpha }^{\beta }{\sqrt {[f(\theta )]^{2}+\left[f'(\theta )\right]^{2}}}\,{\text{d}}\theta \ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd32c346f9d4bd84f08a3f98d6bc826f5f7b28f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.508ex; height:6.343ex;" alt="{\displaystyle s=\int _{\alpha }^{\beta }{\sqrt {[f(\theta )]^{2}+\left[f'(\theta )\right]^{2}}}\,{\text{d}}\theta \ }"></span> </p> </blockquote> <p>En la mayoría de los casos, no hay una solución cerrada disponible y será necesario usar métodos de integración numérica. Por ejemplo, aplicar esta fórmula a la circunferencia de una elipse llevará a una <a href="/wiki/Integral_el%C3%ADptica_de_segunda_especie" class="mw-redirect" title="Integral elíptica de segunda especie">integral elíptica de segunda especie</a>. Entre las curvas con soluciones cerradas están la <a href="/wiki/Catenaria" title="Catenaria">catenaria</a>, el <a href="/wiki/C%C3%ADrculo" title="Círculo">círculo</a>, la <a href="/wiki/Cicloide" title="Cicloide">cicloide</a>, la <a href="/wiki/Espiral_logar%C3%ADtmica" title="Espiral logarítmica">espiral logarítmica</a>, la <a href="/wiki/Par%C3%A1bola_(matem%C3%A1tica)" title="Parábola (matemática)">parábola</a>, la <a href="/wiki/Par%C3%A1bola_semic%C3%BAbica" title="Parábola semicúbica">parábola semicúbica</a> y la línea <a href="/wiki/Recta" title="Recta">recta</a>. </p><p>Un caso un poco más general que el último, es el caso de <a href="/wiki/Coordenadas_curvil%C3%ADneas" title="Coordenadas curvilíneas">coordenadas curvilíneas</a> generales (e incluso el de espacios no euclídeos) caracterizadas por un <a href="/wiki/Tensor_m%C3%A9trico" title="Tensor métrico">tensor métrico</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{ik}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{ik}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9565b09abbe39ff7983a3c925236fd4606e097f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.765ex; height:2.009ex;" alt="{\displaystyle g_{ik}}"></span> donde la longitud de una curva <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C:[a,b]\to M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>:</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C:[a,b]\to M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e39c2b416fc0e42738a63a7b5c648be285687021" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.315ex; height:2.843ex;" alt="{\displaystyle C:[a,b]\to M}"></span> viene dada por: </p> <blockquote style="padding: 5px 10px; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align: left; margin-left:30px; margin-bottom: 0.4em; margin-top:0.2em; min-width:50%;"> <p><span style="float: right; width: 10%; text-align: right;">(<cite id="Equation_4" style="font-style: normal;"><a href="#Eqnref_4">4</a></cite>)</span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=\int _{a}^{b}{\sqrt {\sum _{i,k}g_{ik}{\frac {{\text{d}}x^{i}}{{\text{d}}t}}{\frac {{\text{d}}x^{k}}{{\text{d}}t}}}}\ {\text{d}}t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>k</mi> </mrow> </munder> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> </msqrt> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=\int _{a}^{b}{\sqrt {\sum _{i,k}g_{ik}{\frac {{\text{d}}x^{i}}{{\text{d}}t}}{\frac {{\text{d}}x^{k}}{{\text{d}}t}}}}\ {\text{d}}t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fdca0a0137f15b27d3ef88cb40a4571fe377b3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:28.457ex; height:7.676ex;" alt="{\displaystyle s=\int _{a}^{b}{\sqrt {\sum _{i,k}g_{ik}{\frac {{\text{d}}x^{i}}{{\text{d}}t}}{\frac {{\text{d}}x^{k}}{{\text{d}}t}}}}\ {\text{d}}t}"></span> </p> </blockquote> <p>Por ejemplo el caso de coordenadas polares se obtiene de este haciendo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{1}=r,x^{2}=\theta ;g_{11}=1,g_{22}=r^{2},g_{12}=g_{21}=0;r=f(\theta ),t=\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mi>r</mi> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>θ</mi> <mo>;</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>;</mo> <mi>r</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>t</mi> <mo>=</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{1}=r,x^{2}=\theta ;g_{11}=1,g_{22}=r^{2},g_{12}=g_{21}=0;r=f(\theta ),t=\theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e0d44d3d1ddbd5047735c71d1661fdb0dff81e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:61.424ex; height:3.176ex;" alt="{\displaystyle x^{1}=r,x^{2}=\theta ;g_{11}=1,g_{22}=r^{2},g_{12}=g_{21}=0;r=f(\theta ),t=\theta }"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Ejemplos_de_cálculo"><span id="Ejemplos_de_c.C3.A1lculo"></span>Ejemplos de cálculo</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Longitud_de_arco&action=edit&section=2" title="Editar sección: Ejemplos de cálculo"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>El perímetro de una circunferencia de radio <i>R</i> puede calcularse a partir de la ecuación de esta curva en coordenadas polares </p> <blockquote style="padding: 5px 10px; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align: left; margin-left:30px; margin-bottom: 0.4em; margin-top:0.2em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\theta )=R,\quad 0\leq \theta \leq 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>R</mi> <mo>,</mo> <mspace width="1em" /> <mn>0</mn> <mo>≤</mo> <mi>θ</mi> <mo>≤</mo> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\theta )=R,\quad 0\leq \theta \leq 2\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6a35e7ec744fce8d36d73adaa189d2171f26b34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.342ex; height:2.843ex;" alt="{\displaystyle f(\theta )=R,\quad 0\leq \theta \leq 2\pi }"></span> </p> </blockquote> <p>Para calcular el perímetro se utiliza entonces la ecuación (<span id="Eqnref_3" class="plainlinks neverexpand"><a class="external text" href="https://es.wikipedia.org/wiki/Longitud_de_arco#Equation_3">3</a></span>) </p> <blockquote style="padding: 5px 10px; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align: left; margin-left:30px; margin-bottom: 0.4em; margin-top:0.2em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=\int _{0}^{2\pi }{\sqrt {[f(\theta )]^{2}+\left[f'(\theta )\right]^{2}}}\,{\text{d}}\theta \ =R\int _{0}^{2\pi }\,{\text{d}}\theta =2\pi R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>[</mo> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>θ</mi> <mtext> </mtext> <mo>=</mo> <mi>R</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>θ</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=\int _{0}^{2\pi }{\sqrt {[f(\theta )]^{2}+\left[f'(\theta )\right]^{2}}}\,{\text{d}}\theta \ =R\int _{0}^{2\pi }\,{\text{d}}\theta =2\pi R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/960c2ce2410420532e6f37e33afc722e9fcaadb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:51.554ex; height:6.176ex;" alt="{\displaystyle s=\int _{0}^{2\pi }{\sqrt {[f(\theta )]^{2}+\left[f'(\theta )\right]^{2}}}\,{\text{d}}\theta \ =R\int _{0}^{2\pi }\,{\text{d}}\theta =2\pi R}"></span> </p> </blockquote> <p>Se obtiene que el perímetro de una circunferencia es proporcional al diámetro, lo que se corresponde con la definición de <a href="/wiki/N%C3%BAmero_%CF%80" title="Número π">pi</a>. </p><p>Para determinar la longitud de un arco de circunferencia, basta restringir el ángulo de barrido de la curva a un intervalo más pequeño. </p> <blockquote style="padding: 5px 10px; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align: left; margin-left:30px; margin-bottom: 0.4em; margin-top:0.2em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\theta )=R,\quad \theta _{0}\leq \theta \leq \theta _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>R</mi> <mo>,</mo> <mspace width="1em" /> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>≤</mo> <mi>θ</mi> <mo>≤</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\theta )=R,\quad \theta _{0}\leq \theta \leq \theta _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a1b86a43fa37451e04eebe538cdb99c269ed1e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.974ex; height:2.843ex;" alt="{\displaystyle f(\theta )=R,\quad \theta _{0}\leq \theta \leq \theta _{1}}"></span> </p> </blockquote> <p>La longitud del arco queda </p> <blockquote style="padding: 5px 10px; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align: left; margin-left:30px; margin-bottom: 0.4em; margin-top:0.2em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=\int _{\theta _{0}}^{\theta _{1}}{\sqrt {[f(\theta )]^{2}+\left[f'(\theta )\right]^{2}}}\,{\text{d}}\theta \ =R\cdot (\theta _{1}-\theta _{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>[</mo> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>θ</mi> <mtext> </mtext> <mo>=</mo> <mi>R</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=\int _{\theta _{0}}^{\theta _{1}}{\sqrt {[f(\theta )]^{2}+\left[f'(\theta )\right]^{2}}}\,{\text{d}}\theta \ =R\cdot (\theta _{1}-\theta _{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/123062c8596848519fffb029965989d8b317ab82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:46.65ex; height:6.676ex;" alt="{\displaystyle s=\int _{\theta _{0}}^{\theta _{1}}{\sqrt {[f(\theta )]^{2}+\left[f'(\theta )\right]^{2}}}\,{\text{d}}\theta \ =R\cdot (\theta _{1}-\theta _{0})}"></span> </p> </blockquote> <div class="mw-heading mw-heading2"><h2 id="Deducción_de_la_fórmula_para_funciones_de_una_variable"><span id="Deducci.C3.B3n_de_la_f.C3.B3rmula_para_funciones_de_una_variable"></span>Deducción de la fórmula para funciones de una variable</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Longitud_de_arco&action=edit&section=3" title="Editar sección: Deducción de la fórmula para funciones de una variable"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Archivo:Arclength.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Arclength.svg/350px-Arclength.svg.png" decoding="async" width="350" height="91" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Arclength.svg/525px-Arclength.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Arclength.svg/700px-Arclength.svg.png 2x" data-file-width="582" data-file-height="152" /></a><figcaption>Aproximación por múltiples segmentos lineales.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Archivo:Arc_length_approximation.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Arc_length_approximation.svg/langes-220px-Arc_length_approximation.svg.png" decoding="async" width="220" height="204" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Arc_length_approximation.svg/langes-330px-Arc_length_approximation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Arc_length_approximation.svg/langes-440px-Arc_length_approximation.svg.png 2x" data-file-width="400" data-file-height="370" /></a><figcaption>Para un pequeño segmento de curva, Δs se puede aproximar con el teorema de Pitágoras.</figcaption></figure> <p>Suponiendo que se tiene una curva <a href="/wiki/Conjunto_rectificable" title="Conjunto rectificable">rectificable</a> cualquiera, determinada por una función <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\left(x\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\left(x\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/653b89efce2f12f2c8bb8a5536ac569fe73e8271" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.805ex; height:2.843ex;" alt="{\displaystyle f\left(x\right)}"></span>, y suponiendo que se quiere aproximar la longitud del arco de curva <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> que va desde un punto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> a uno <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>. Con este propósito es posible diseñar una serie de triángulos rectángulos cuyas hipotenusas concatenadas "cubran" el arco de curva elegido tal como se ve en la figura. Para hacer a este método "más funcional" también se puede exigir que las bases de todos aquellos triángulos sean iguales a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3890eb866b6258d7a304fc34c70ee3fb3a81a70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.266ex; height:2.176ex;" alt="{\displaystyle \Delta x}"></span>, de manera que para cada uno existirá un cateto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7caae142d915be8ef4d8c423bf91d1f6ea10e8e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.091ex; height:2.509ex;" alt="{\displaystyle \Delta y}"></span> asociado, dependiendo del tipo de curva y del arco elegido, siendo entonces cada hipotenusa, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta s={\sqrt {\Delta x^{2}+\Delta y^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi mathvariant="normal">Δ</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta s={\sqrt {\Delta x^{2}+\Delta y^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/420cbabb0824b2df59a534c2c83f2204534e3263" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:19.759ex; height:4.843ex;" alt="{\displaystyle \Delta s={\sqrt {\Delta x^{2}+\Delta y^{2}}}}"></span>, al aplicarse el <a href="/wiki/Teorema_de_Pit%C3%A1goras" title="Teorema de Pitágoras">teorema de Pitágoras</a>. Así, una aproximación de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> estaría dada por la sumatoria de todas aquellas <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> hipotenusas desplegadas. Por eso se tiene que: </p> <blockquote style="padding: 5px 10px; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align: left; margin-left:30px; margin-bottom: 0.4em; margin-top:0.2em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\approx \sum _{i=1}^{n}{\sqrt {\Delta x_{i}^{2}+\Delta y_{i}^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>≈</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi mathvariant="normal">Δ</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s\approx \sum _{i=1}^{n}{\sqrt {\Delta x_{i}^{2}+\Delta y_{i}^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9a6f317a5206ee5de5969bb93a605418145847e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.566ex; height:6.843ex;" alt="{\displaystyle s\approx \sum _{i=1}^{n}{\sqrt {\Delta x_{i}^{2}+\Delta y_{i}^{2}}}}"></span> </p> </blockquote> <p>Pasando a operar algebraicamente la forma en la que se calcula cada hipotenusa para llegar a una nueva expresión; </p> <blockquote style="padding: 5px 10px; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align: left; margin-left:30px; margin-bottom: 0.4em; margin-top:0.2em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\Delta x^{2}+\Delta y^{2}}}={\sqrt {({\Delta x^{2}+\Delta y^{2}})\left({\frac {\Delta x^{2}}{\Delta x^{2}}}\right)}}={\sqrt {1+\left({\frac {\Delta y^{2}}{\Delta x^{2}}}\right)}}\Delta x={\sqrt {1+\left({\frac {\Delta y}{\Delta x}}\right)^{2}}}\Delta x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi mathvariant="normal">Δ</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </msqrt> </mrow> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\Delta x^{2}+\Delta y^{2}}}={\sqrt {({\Delta x^{2}+\Delta y^{2}})\left({\frac {\Delta x^{2}}{\Delta x^{2}}}\right)}}={\sqrt {1+\left({\frac {\Delta y^{2}}{\Delta x^{2}}}\right)}}\Delta x={\sqrt {1+\left({\frac {\Delta y}{\Delta x}}\right)^{2}}}\Delta x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83fb12307576aa76f99fafa7795846b406f8098d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:83.677ex; height:7.843ex;" alt="{\displaystyle {\sqrt {\Delta x^{2}+\Delta y^{2}}}={\sqrt {({\Delta x^{2}+\Delta y^{2}})\left({\frac {\Delta x^{2}}{\Delta x^{2}}}\right)}}={\sqrt {1+\left({\frac {\Delta y^{2}}{\Delta x^{2}}}\right)}}\Delta x={\sqrt {1+\left({\frac {\Delta y}{\Delta x}}\right)^{2}}}\Delta x}"></span> </p> </blockquote> <p>Luego, el resultado previo toma la siguiente forma: </p> <blockquote style="padding: 5px 10px; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align: left; margin-left:30px; margin-bottom: 0.4em; margin-top:0.2em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\approx \sum _{i=1}^{n}{\sqrt {1+\left({\frac {\Delta y_{i}}{\Delta x_{i}}}\right)^{2}}}\Delta x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>≈</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mi mathvariant="normal">Δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s\approx \sum _{i=1}^{n}{\sqrt {1+\left({\frac {\Delta y_{i}}{\Delta x_{i}}}\right)^{2}}}\Delta x_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52790f873ffebef8301c78c00cabd6d486c54d1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.7ex; height:7.676ex;" alt="{\displaystyle s\approx \sum _{i=1}^{n}{\sqrt {1+\left({\frac {\Delta y_{i}}{\Delta x_{i}}}\right)^{2}}}\Delta x_{i}}"></span> </p> </blockquote> <p>Ahora bien, mientras más pequeños sean estos <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> segmentos, mejor será la aproximación buscada; serán tan pequeños como se desee, de modo que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3890eb866b6258d7a304fc34c70ee3fb3a81a70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.266ex; height:2.176ex;" alt="{\displaystyle \Delta x}"></span> tienda a cero. Así, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3890eb866b6258d7a304fc34c70ee3fb3a81a70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.266ex; height:2.176ex;" alt="{\displaystyle \Delta x}"></span> se convierte en <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/845c817e348381a13f3fad5184169ce0e021c685" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.546ex; height:2.176ex;" alt="{\displaystyle dx}"></span>, y cada cociente incremental <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\Delta y_{i}/\Delta x_{i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">Δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Delta y_{i}/\Delta x_{i}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/154f647d5ab4aff51efb41f19805797e22a71956" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.102ex; height:2.843ex;" alt="{\displaystyle {\Delta y_{i}/\Delta x_{i}}}"></span> se transforma en un <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dy/dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dy/dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add69069028cada9be6b945dd4b9895e3ff2fd23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.079ex; height:2.843ex;" alt="{\displaystyle dy/dx}"></span> general, que es por definición <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'\left(x\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'\left(x\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e0e6240e3512deb25157c701ee38a1e1262b36a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.531ex; height:3.009ex;" alt="{\displaystyle f'\left(x\right)}"></span>. Dados estos cambios, la aproximación anterior se convierte en una sumatoria más fina y ahora exacta, una integración de infinitos segmentos infinitesimales; </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=\lim _{\Delta x_{i}\to 0}\sum _{i=1}^{\infty }{\sqrt {1+\left({\frac {\Delta y_{i}}{\Delta x_{i}}}\right)^{2}}}\Delta x_{i}=\int _{a}^{b}{\sqrt {1+\left({\frac {{\text{d}}y}{{\text{d}}x}}\right)^{2}}}{\text{d}}x=\int _{a}^{b}{\sqrt {1+\left[f'\left(x\right)\right]^{2}}}\,{\text{d}}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mi mathvariant="normal">Δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mo>[</mo> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=\lim _{\Delta x_{i}\to 0}\sum _{i=1}^{\infty }{\sqrt {1+\left({\frac {\Delta y_{i}}{\Delta x_{i}}}\right)^{2}}}\Delta x_{i}=\int _{a}^{b}{\sqrt {1+\left({\frac {{\text{d}}y}{{\text{d}}x}}\right)^{2}}}{\text{d}}x=\int _{a}^{b}{\sqrt {1+\left[f'\left(x\right)\right]^{2}}}\,{\text{d}}x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09e509a9bb943b6e6df6f9085267216089683554" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:81.358ex; height:7.676ex;" alt="{\displaystyle s=\lim _{\Delta x_{i}\to 0}\sum _{i=1}^{\infty }{\sqrt {1+\left({\frac {\Delta y_{i}}{\Delta x_{i}}}\right)^{2}}}\Delta x_{i}=\int _{a}^{b}{\sqrt {1+\left({\frac {{\text{d}}y}{{\text{d}}x}}\right)^{2}}}{\text{d}}x=\int _{a}^{b}{\sqrt {1+\left[f'\left(x\right)\right]^{2}}}\,{\text{d}}x}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Métodos_anteriores_al_cálculo"><span id="M.C3.A9todos_anteriores_al_c.C3.A1lculo"></span>Métodos anteriores al cálculo</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Longitud_de_arco&action=edit&section=4" title="Editar sección: Métodos anteriores al cálculo"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Antigüedad"><span id="Antig.C3.BCedad"></span>Antigüedad</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Longitud_de_arco&action=edit&section=5" title="Editar sección: Antigüedad"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A través de la historia de las matemáticas, grandes pensadores consideraron imposible calcular la longitud de un arco irregular. Aunque <a href="/wiki/Arqu%C3%ADmedes" title="Arquímedes">Arquímedes</a> había descubierto una <a href="/w/index.php?title=Aproximaci%C3%B3n_rectangular&action=edit&redlink=1" class="new" title="Aproximación rectangular (aún no redactado)">aproximación rectangular</a> para calcular el área bajo una curva con un <a href="/wiki/M%C3%A9todo_de_agotamiento" class="mw-redirect" title="Método de agotamiento">método de agotamiento</a>, pocos creyeron que fuera posible que una curva tuviese una longitud definida, como las líneas rectas. Las primeras mediciones se hicieron posibles, como ya es común en el cálculo, a través de aproximaciones: los matemáticos de la época trazaban un <a href="/wiki/Pol%C3%ADgono" title="Polígono">polígono</a> dentro de la curva, y calculaban la longitud de los lados de éste para obtener un valor aproximado de la longitud de la curva. Mientras se usaban más segmentos, disminuyendo la longitud de cada uno, se obtenía una aproximación cada vez mejor. </p> <div class="mw-heading mw-heading3"><h3 id="Siglo_XVII">Siglo <span style="font-variant:small-caps;text-transform:lowercase">XVII</span></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Longitud_de_arco&action=edit&section=6" title="Editar sección: Siglo XVII"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>En esta época, el método de agotamiento llevó a la rectificación por métodos geométricos de muchas curvas trascendentales: la <a href="/wiki/Espiral_logar%C3%ADtmica" title="Espiral logarítmica">Espiral logarítmica</a> de <a href="/wiki/Evangelista_Torricelli" title="Evangelista Torricelli">Torricelli</a> en 1645 (algunos piensan que fue <a href="/wiki/John_Wallis" title="John Wallis">John Wallis</a> en 1650), el <a href="/wiki/Cicloide" title="Cicloide">Cicloide</a> de <a href="/wiki/Christopher_Wren" title="Christopher Wren">Christopher Wren</a> en 1658, y la <a href="/wiki/Catenaria" title="Catenaria">Catenaria</a> de <a href="/wiki/Gottfried_Leibniz" title="Gottfried Leibniz">Gottfried Leibniz</a> en 1691. </p> <div class="mw-heading mw-heading2"><h2 id="Véase_también"><span id="V.C3.A9ase_tambi.C3.A9n"></span>Véase también</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Longitud_de_arco&action=edit&section=7" title="Editar sección: Véase también"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Curva#Geometría_diferencial_de_curvas_en_R3" title="Curva">Geometría diferencial de curvas en ℝ³</a></li> <li><a href="/wiki/Arco_(geometr%C3%ADa)" title="Arco (geometría)">Arco (geometría)</a></li> <li><a href="/wiki/Integraci%C3%B3n_num%C3%A9rica" title="Integración numérica">Integración numérica</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Enlaces_externos">Enlaces externos</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Longitud_de_arco&action=edit&section=8" title="Editar sección: Enlaces externos"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20051219223123/http://math.kennesaw.edu/~jdoto/13.pdf">Math Before Calculus</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20071106083431/http://www3.villanova.edu/maple/misc/history_of_curvature/k.htm">The History of Curvature</a></li> <li><span id="Reference-Mathworld-Longitud_de_arco" class="citation web"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W</a>. <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/ArcLength.html">«Longitud de arco»</a>. En Weisstein, Eric W, ed. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i> <span style="color:var(--color-subtle, #555 );">(en inglés)</span>. <a href="/wiki/Wolfram_Research" title="Wolfram Research">Wolfram Research</a>.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3ALongitud+de+arco&rft.atitle=Longitud+de+arco&rft.au=Weisstein%2C+Eric+W&rft.aulast=Weisstein%2C+Eric+W&rft.genre=article&rft.jtitle=MathWorld&rft.pub=Wolfram+Research&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FArcLength.html&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></li> <li><span id="CITAREFEd_Pegg,_Jr." class="citation web">Ed Pegg, Jr. <a rel="nofollow" class="external text" href="http://demonstrations.wolfram.com/ArcLength/">«Longitud de arco»</a>. <i><a href="/wiki/Wolfram_Demonstrations_Project" title="Wolfram Demonstrations Project">The Wolfram Demonstrations Project</a></i> <span style="color:var(--color-subtle, #555 );">(en inglés)</span>. <a href="/wiki/Wolfram_Research" title="Wolfram Research">Wolfram Research</a>.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3ALongitud+de+arco&rft.atitle=Longitud+de+arco&rft.au=Ed+Pegg%2C+Jr.&rft.aulast=Ed+Pegg%2C+Jr.&rft.genre=article&rft.jtitle=The+Wolfram+Demonstrations+Project&rft.pub=Wolfram+Research&rft_id=http%3A%2F%2Fdemonstrations.wolfram.com%2FArcLength%2F&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></li> <li><a rel="nofollow" class="external text" href="http://www.pinkmonkey.com/studyguides/subjects/calc/chap8/c0808501.asp">Calculus Study Guide – Arc Length (Rectification)</a></li> <li><a rel="nofollow" class="external text" href="http://www-groups.dcs.st-and.ac.uk/~history/Curves/Curves.html">Famous Curves Index</a> <i>The MacTutor History of Mathematics archive</i></li> <li><span id="CITAREFChad_Pierson,_Josh_Fritz,_Angela_Sharp" class="citation web">Chad Pierson, Josh Fritz, Angela Sharp. <a rel="nofollow" class="external text" href="http://demonstrations.wolfram.com/ArcLengthApproximation/">«Aproximación de la longitud de arco»</a>. <i><a href="/wiki/Wolfram_Demonstrations_Project" title="Wolfram Demonstrations Project">The Wolfram Demonstrations Project</a></i> <span style="color:var(--color-subtle, #555 );">(en inglés)</span>. <a href="/wiki/Wolfram_Research" title="Wolfram Research">Wolfram Research</a>.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3ALongitud+de+arco&rft.atitle=Aproximaci%C3%B3n+de+la+longitud+de+arco&rft.au=Chad+Pierson%2C+Josh+Fritz%2C+Angela+Sharp&rft.aulast=Chad+Pierson%2C+Josh+Fritz%2C+Angela+Sharp&rft.genre=article&rft.jtitle=The+Wolfram+Demonstrations+Project&rft.pub=Wolfram+Research&rft_id=http%3A%2F%2Fdemonstrations.wolfram.com%2FArcLengthApproximation%2F&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></li> <li><a rel="nofollow" class="external text" href="http://numericalmethods.eng.usf.edu/experiments/Length_of_curve_experiment.pdf">Length of a Curve Experiment</a> Illustrates numerical solution of finding length of a curve.</li></ul> <style data-mw-deduplicate="TemplateStyles:r161257576">.mw-parser-output .mw-authority-control{margin-top:1.5em}.mw-parser-output .mw-authority-control .navbox table{margin:0}.mw-parser-output .mw-authority-control .navbox 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style="white-space:nowrap;"><span typeof="mw:File"><a href="/wiki/Wikidata" title="Wikidata"><img alt="Wd" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/20px-Wikidata-logo.svg.png" decoding="async" width="20" height="11" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/30px-Wikidata-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/40px-Wikidata-logo.svg.png 2x" data-file-width="1050" data-file-height="590" /></a></span> Datos:</span> <span class="uid"><a href="https://www.wikidata.org/wiki/Q670036" class="extiw" title="wikidata:Q670036">Q670036</a></span></li> <li><span style="white-space:nowrap;"><span typeof="mw:File"><a href="/wiki/Wikimedia_Commons" title="Commonscat"><img alt="Commonscat" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/15px-Commons-logo.svg.png" decoding="async" width="15" height="20" class="mw-file-element" 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style="white-space:nowrap;"><span typeof="mw:File"><a href="/wiki/Wikidata" title="Wikidata"><img alt="Wd" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/20px-Wikidata-logo.svg.png" decoding="async" width="20" height="11" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/30px-Wikidata-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/40px-Wikidata-logo.svg.png 2x" data-file-width="1050" data-file-height="590" /></a></span> Datos:</span> <span class="uid"><a href="https://www.wikidata.org/wiki/Q670036" class="extiw" title="wikidata:Q670036">Q670036</a></span></li> <li><span style="white-space:nowrap;"><span typeof="mw:File"><a href="/wiki/Wikimedia_Commons" title="Commonscat"><img alt="Commonscat" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/15px-Commons-logo.svg.png" decoding="async" width="15" height="20" class="mw-file-element" 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Longitud de arco - Wikipedia, la enciclopedia libre