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data-event-name="pinnable-header.vector-toc.unpin">Escondre</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Començament</div> </a> </li> <li id="toc-Unitats" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Unitats"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Unitats</span> </div> </a> <ul id="toc-Unitats-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fórmulas_per_calcular_de_volums" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Fórmulas_per_calcular_de_volums"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Fórmulas per calcular de volums</span> </div> </a> <button aria-controls="toc-Fórmulas_per_calcular_de_volums-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>Commuta la subsecció Fórmulas per calcular de volums</span> </button> <ul id="toc-Fórmulas_per_calcular_de_volums-sublist" class="vector-toc-list"> <li id="toc-Volum_dels_solids_platonics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Volum_dels_solids_platonics"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Volum dels solids platonics</span> </div> </a> <ul id="toc-Volum_dels_solids_platonics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Proporcion_entre_los_volums_d'un_còn,_esfèra_e_cilindre_del_meteis_radi_e_nautor" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Proporcion_entre_los_volums_d'un_còn,_esfèra_e_cilindre_del_meteis_radi_e_nautor"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Proporcion entre los volums d'un còn, esfèra e cilindre del meteis radi e nautor</span> </div> </a> <ul id="toc-Proporcion_entre_los_volums_d'un_còn,_esfèra_e_cilindre_del_meteis_radi_e_nautor-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Calcul_del_volum_per_integralas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Calcul_del_volum_per_integralas"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Calcul del volum per integralas</span> </div> </a> <button aria-controls="toc-Calcul_del_volum_per_integralas-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>Commuta la subsecció Calcul del volum per integralas</span> </button> <ul id="toc-Calcul_del_volum_per_integralas-sublist" class="vector-toc-list"> <li id="toc-Coordenadas_non_cartesianas" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Coordenadas_non_cartesianas"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Coordenadas non cartesianas</span> </div> </a> <ul id="toc-Coordenadas_non_cartesianas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Solid_de_rotacion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Solid_de_rotacion"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Solid de rotacion</span> </div> </a> <ul id="toc-Solid_de_rotacion-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Teorèma_de_la_divergéncia" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Teorèma_de_la_divergéncia"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Teorèma de la divergéncia</span> </div> </a> <ul id="toc-Teorèma_de_la_divergéncia-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Referéncias" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referéncias"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Referéncias</span> </div> </a> <ul id="toc-Referéncias-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vejatz_tanben" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vejatz_tanben"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Vejatz tanben</span> </div> </a> <ul id="toc-Vejatz_tanben-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ligams_extèrnes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ligams_extèrnes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Ligams extèrnes</span> </div> </a> <ul id="toc-Ligams_extèrnes-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="Somari" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Taula de continguts" > <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="Commuta la taula de continguts." > <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">Commuta la taula de continguts.</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">Volum</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="Vés a un article en una altra llengua. Disponible en 120 llengües" > <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-120" 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">120 lengas</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Volume" title="Volume - afrikaans" lang="af" hreflang="af" data-title="Volume" data-language-autonym="Afrikaans" data-language-local-name="afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-anp mw-list-item"><a href="https://anp.wikipedia.org/wiki/%E0%A4%86%E0%A4%AF%E0%A4%A4%E0%A4%A8" title="आयतन - angika" lang="anp" hreflang="anp" data-title="आयतन" data-language-autonym="अंगिका" data-language-local-name="angika" class="interlanguage-link-target"><span>अंगिका</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AD%D8%AC%D9%85" title="حجم - arabi" lang="ar" hreflang="ar" data-title="حجم" data-language-autonym="العربية" data-language-local-name="arabi" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://ary.wikipedia.org/wiki/%D9%82%D8%AF" title="قد - arabe marocain" lang="ary" hreflang="ary" data-title="قد" data-language-autonym="الدارجة" data-language-local-name="arabe marocain" class="interlanguage-link-target"><span>الدارجة</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Volume" title="Volume - asturian" lang="ast" hreflang="ast" data-title="Volume" data-language-autonym="Asturianu" data-language-local-name="asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/H%C9%99cm" title="Həcm - azerbaijani" lang="az" hreflang="az" data-title="Həcm" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9A%D2%AF%D0%BB%D3%99%D0%BC" title="Күләм - baixkir" lang="ba" hreflang="ba" data-title="Күләм" data-language-autonym="Башҡортса" data-language-local-name="baixkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Bolyum" title="Bolyum - Central Bikol" lang="bcl" hreflang="bcl" data-title="Bolyum" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target"><span>Bikol Central</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%90%D0%B1%E2%80%99%D1%91%D0%BC" title="Аб’ём - belarús" lang="be" hreflang="be" data-title="Аб’ём" data-language-autonym="Беларуская" data-language-local-name="belarús" 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%90%D0%B1%E2%80%99%D1%91%D0%BC" title="Аб’ём - Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Аб’ём" 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-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9E%D0%B1%D0%B5%D0%BC" title="Обем - bulgar" lang="bg" hreflang="bg" data-title="Обем" data-language-autonym="Български" data-language-local-name="bulgar" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%86%E0%A4%AF%E0%A4%A4%E0%A4%A8" title="आयतन - Bhojpuri" lang="bh" hreflang="bh" data-title="आयतन" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target"><span>भोजपुरी</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%86%E0%A6%AF%E0%A6%BC%E0%A6%A4%E0%A6%A8" title="আয়তন - bengalin" lang="bn" hreflang="bn" data-title="আয়তন" data-language-autonym="বাংলা" data-language-local-name="bengalin" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Volum" title="Volum - breton" lang="br" hreflang="br" data-title="Volum" data-language-autonym="Brezhoneg" data-language-local-name="breton" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Zapremina" title="Zapremina - bosniac" lang="bs" hreflang="bs" data-title="Zapremina" data-language-autonym="Bosanski" data-language-local-name="bosniac" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Volum" title="Volum - catalan" lang="ca" hreflang="ca" data-title="Volum" data-language-autonym="Català" data-language-local-name="catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cdo mw-list-item"><a href="https://cdo.wikipedia.org/wiki/T%C4%81%CC%A4-c%C3%A9k" title="Tā̤-cék - Mindong" lang="cdo" hreflang="cdo" data-title="Tā̤-cék" data-language-autonym="閩東語 / Mìng-dĕ̤ng-ngṳ̄" data-language-local-name="Mindong" class="interlanguage-link-target"><span>閩東語 / Mìng-dĕ̤ng-ngṳ̄</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%82%DB%95%D8%A8%D8%A7%D8%B1%DB%95" title="قەبارە - kurd central" lang="ckb" hreflang="ckb" data-title="قەبارە" data-language-autonym="کوردی" data-language-local-name="kurd central" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Objem" title="Objem - chèc" lang="cs" hreflang="cs" data-title="Objem" data-language-autonym="Čeština" data-language-local-name="chèc" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9A%D0%B0%D0%BB%C4%83%D0%BF%C4%83%D1%88" title="Калăпăш - chovash" lang="cv" hreflang="cv" data-title="Калăпăш" data-language-autonym="Чӑвашла" data-language-local-name="chovash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Cyfaint" title="Cyfaint - gal·lès" lang="cy" hreflang="cy" data-title="Cyfaint" data-language-autonym="Cymraeg" data-language-local-name="gal·lès" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Rumfang" title="Rumfang - danés" lang="da" hreflang="da" data-title="Rumfang" 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/Volumen" title="Volumen - alemand" lang="de" hreflang="de" data-title="Volumen" data-language-autonym="Deutsch" data-language-local-name="alemand" 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%8C%CE%B3%CE%BA%CE%BF%CF%82" title="Όγκος - grèc" lang="el" hreflang="el" data-title="Όγκος" data-language-autonym="Ελληνικά" data-language-local-name="grèc" 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/Volume" title="Volume - anglés" lang="en" hreflang="en" data-title="Volume" data-language-autonym="English" data-language-local-name="anglés" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Volumeno" title="Volumeno - esperanto" lang="eo" hreflang="eo" data-title="Volumeno" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Volumen" title="Volumen - espanhòl" lang="es" hreflang="es" data-title="Volumen" data-language-autonym="Español" data-language-local-name="espanhòl" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Ruumala" title="Ruumala - estonian" lang="et" hreflang="et" data-title="Ruumala" data-language-autonym="Eesti" data-language-local-name="estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Bolumen_(espazioa)" title="Bolumen (espazioa) - basc" lang="eu" hreflang="eu" data-title="Bolumen (espazioa)" data-language-autonym="Euskara" data-language-local-name="basc" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AD%D8%AC%D9%85" title="حجم - perse" lang="fa" hreflang="fa" data-title="حجم" data-language-autonym="فارسی" data-language-local-name="perse" 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/Tilavuus" title="Tilavuus - finlandés" lang="fi" hreflang="fi" data-title="Tilavuus" data-language-autonym="Suomi" data-language-local-name="finlandés" 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/Volume" title="Volume - francés" lang="fr" hreflang="fr" data-title="Volume" data-language-autonym="Français" data-language-local-name="francés" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/R%C3%BCm_(Miat)" title="Rüm (Miat) - frisó septentrional" lang="frr" hreflang="frr" data-title="Rüm (Miat)" data-language-autonym="Nordfriisk" data-language-local-name="frisó septentrional" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Toirt" title="Toirt - irlandés" lang="ga" hreflang="ga" data-title="Toirt" data-language-autonym="Gaeilge" data-language-local-name="irlandés" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E9%AB%94%E7%A9%8D" title="體積 - xinès gan" lang="gan" hreflang="gan" data-title="體積" data-language-autonym="贛語" data-language-local-name="xinès gan" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Voloum" title="Voloum - créole guyanais" lang="gcr" hreflang="gcr" data-title="Voloum" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="créole guyanais" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Volume_(magnitude)" title="Volume (magnitude) - galician" lang="gl" hreflang="gl" data-title="Volume (magnitude)" data-language-autonym="Galego" data-language-local-name="galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-hak mw-list-item"><a href="https://hak.wikipedia.org/wiki/Th%C3%AD-chit" title="Thí-chit - xinès hakka" lang="hak" hreflang="hak" data-title="Thí-chit" data-language-autonym="客家語 / Hak-kâ-ngî" data-language-local-name="xinès hakka" class="interlanguage-link-target"><span>客家語 / Hak-kâ-ngî</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A0%D7%A4%D7%97" title="נפח - ebrèu" lang="he" hreflang="he" data-title="נפח" data-language-autonym="עברית" data-language-local-name="ebrèu" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%86%E0%A4%AF%E0%A4%A4%E0%A4%A8" title="आयतन - Indi" lang="hi" hreflang="hi" data-title="आयतन" data-language-autonym="हिन्दी" data-language-local-name="Indi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Pasaar" title="Pasaar - hindi de Fiji" lang="hif" hreflang="hif" data-title="Pasaar" data-language-autonym="Fiji Hindi" data-language-local-name="hindi de Fiji" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Obujam" title="Obujam - croat" lang="hr" hreflang="hr" data-title="Obujam" data-language-autonym="Hrvatski" data-language-local-name="croat" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Volim" title="Volim - crioll d’Haití" lang="ht" hreflang="ht" data-title="Volim" data-language-autonym="Kreyòl ayisyen" data-language-local-name="crioll d’Haití" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/T%C3%A9rfogat" title="Térfogat - ongrés" lang="hu" hreflang="hu" data-title="Térfogat" data-language-autonym="Magyar" data-language-local-name="ongrés" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BE%D5%A1%D5%BE%D5%A1%D5%AC" title="Ծավալ - armèni" lang="hy" hreflang="hy" data-title="Ծավալ" data-language-autonym="Հայերեն" data-language-local-name="armèni" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Volumine" title="Volumine - interlingua" lang="ia" hreflang="ia" data-title="Volumine" data-language-autonym="Interlingua" data-language-local-name="interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Volume" title="Volume - indonesian" lang="id" hreflang="id" data-title="Volume" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Volumino" title="Volumino - ido" lang="io" hreflang="io" data-title="Volumino" data-language-autonym="Ido" data-language-local-name="ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/R%C3%BAmm%C3%A1l" title="Rúmmál - islandés" lang="is" hreflang="is" data-title="Rúmmál" 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/Volume" title="Volume - italian" lang="it" hreflang="it" data-title="Volume" data-language-autonym="Italiano" data-language-local-name="italian" 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/%E4%BD%93%E7%A9%8D" title="体積 - japonés" lang="ja" hreflang="ja" data-title="体積" 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-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Valyuum" title="Valyuum - crioll anglès de Jamaica" lang="jam" hreflang="jam" data-title="Valyuum" data-language-autonym="Patois" data-language-local-name="crioll anglès de Jamaica" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Volume" title="Volume - javanés" lang="jv" hreflang="jv" data-title="Volume" data-language-autonym="Jawa" data-language-local-name="javanés" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9B%E1%83%9D%E1%83%AA%E1%83%A3%E1%83%9A%E1%83%9D%E1%83%91%E1%83%90" title="მოცულობა - georgian" lang="ka" hreflang="ka" data-title="მოცულობა" data-language-autonym="ქართული" data-language-local-name="georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Able%C9%A3" title="Ableɣ - cabil" lang="kab" hreflang="kab" data-title="Ableɣ" data-language-autonym="Taqbaylit" data-language-local-name="cabil" class="interlanguage-link-target"><span>Taqbaylit</span></a></li><li class="interlanguage-link interwiki-kbp mw-list-item"><a href="https://kbp.wikipedia.org/wiki/Wonuu_pa%C9%A3laka" title="Wonuu paɣlaka - Kabiye" lang="kbp" hreflang="kbp" data-title="Wonuu paɣlaka" data-language-autonym="Kabɩyɛ" data-language-local-name="Kabiye" class="interlanguage-link-target"><span>Kabɩyɛ</span></a></li><li class="interlanguage-link interwiki-kge mw-list-item"><a href="https://kge.wikipedia.org/wiki/Isi_(Volume)" title="Isi (Volume) - Komering" lang="kge" hreflang="kge" data-title="Isi (Volume)" data-language-autonym="Kumoring" data-language-local-name="Komering" class="interlanguage-link-target"><span>Kumoring</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9A%D3%A9%D0%BB%D0%B5%D0%BC" title="Көлем - cazac" lang="kk" hreflang="kk" data-title="Көлем" data-language-autonym="Қазақша" data-language-local-name="cazac" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B6%80%ED%94%BC" title="부피 - corean" lang="ko" hreflang="ko" data-title="부피" data-language-autonym="한국어" data-language-local-name="corean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-krc mw-list-item"><a href="https://krc.wikipedia.org/wiki/%D0%A1%D1%8B%D0%B9%D1%8B%D0%BC" title="Сыйым - karatxai-balkar" lang="krc" hreflang="krc" data-title="Сыйым" data-language-autonym="Къарачай-малкъар" data-language-local-name="karatxai-balkar" class="interlanguage-link-target"><span>Къарачай-малкъар</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Qeware" title="Qeware - curd" lang="ku" hreflang="ku" data-title="Qeware" data-language-autonym="Kurdî" data-language-local-name="curd" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9A%D3%A9%D0%BB%D3%A9%D0%BC" title="Көлөм - kirguís" lang="ky" hreflang="ky" data-title="Көлөм" data-language-autonym="Кыргызча" data-language-local-name="kirguís" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Capacitas_et_volumen" title="Capacitas et volumen - latin" lang="la" hreflang="la" data-title="Capacitas et volumen" data-language-autonym="Latina" data-language-local-name="latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Volumen" title="Volumen - luxemborgés" lang="lb" hreflang="lb" data-title="Volumen" data-language-autonym="Lëtzebuergesch" data-language-local-name="luxemborgés" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lij mw-list-item"><a href="https://lij.wikipedia.org/wiki/Vol%C3%B9mme" title="Volùmme - lígur" lang="lij" hreflang="lij" data-title="Volùmme" data-language-autonym="Ligure" data-language-local-name="lígur" class="interlanguage-link-target"><span>Ligure</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/T%C5%ABris" title="Tūris - lituan" lang="lt" hreflang="lt" data-title="Tūris" data-language-autonym="Lietuvių" data-language-local-name="lituan" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-ltg mw-list-item"><a href="https://ltg.wikipedia.org/wiki/Vydums" title="Vydums - latgalien" lang="ltg" hreflang="ltg" data-title="Vydums" data-language-autonym="Latgaļu" data-language-local-name="latgalien" class="interlanguage-link-target"><span>Latgaļu</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Tilpums" title="Tilpums - leton" lang="lv" hreflang="lv" data-title="Tilpums" data-language-autonym="Latviešu" data-language-local-name="leton" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Hadiry" title="Hadiry - malgash" lang="mg" hreflang="mg" data-title="Hadiry" data-language-autonym="Malagasy" data-language-local-name="malgash" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%97%D0%B0%D1%84%D0%B0%D1%82%D0%BD%D0%B8%D0%BD%D0%B0" title="Зафатнина - macedonian" lang="mk" hreflang="mk" data-title="Зафатнина" data-language-autonym="Македонски" data-language-local-name="macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B5%E0%B5%8D%E0%B4%AF%E0%B4%BE%E0%B4%AA%E0%B5%8D%E0%B4%A4%E0%B4%82" title="വ്യാപ്തം - malaiàlam" lang="ml" hreflang="ml" data-title="വ്യാപ്തം" data-language-autonym="മലയാളം" data-language-local-name="malaiàlam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%AD%D0%B7%D0%BB%D1%8D%D1%85%D2%AF%D2%AF%D0%BD" title="Эзлэхүүн - mongòl" lang="mn" hreflang="mn" data-title="Эзлэхүүн" data-language-autonym="Монгол" data-language-local-name="mongòl" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%86%E0%A4%95%E0%A4%BE%E0%A4%B0%E0%A4%AE%E0%A4%BE%E0%A4%A8" title="आकारमान - marathi" lang="mr" hreflang="mr" data-title="आकारमान" data-language-autonym="मराठी" data-language-local-name="marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Isi_padu" title="Isi padu - malai" lang="ms" hreflang="ms" data-title="Isi padu" data-language-autonym="Bahasa Melayu" data-language-local-name="malai" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%91%E1%80%AF%E1%80%91%E1%80%8A%E1%80%BA" title="ထုထည် - birman" lang="my" hreflang="my" data-title="ထုထည်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="birman" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%86%E0%A4%AF%E0%A4%A4%E0%A4%A8" title="आयतन - nepalés" lang="ne" hreflang="ne" data-title="आयतन" data-language-autonym="नेपाली" data-language-local-name="nepalés" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Inhoud_(volume)" title="Inhoud (volume) - neerlandés" lang="nl" hreflang="nl" data-title="Inhoud (volume)" data-language-autonym="Nederlands" data-language-local-name="neerlandés" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Volum" title="Volum - norvegian nynorsk" lang="nn" hreflang="nn" data-title="Volum" data-language-autonym="Norsk nynorsk" data-language-local-name="norvegian nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Volum" title="Volum - norvegian bokmål" lang="nb" hreflang="nb" data-title="Volum" data-language-autonym="Norsk bokmål" data-language-local-name="norvegian bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Qabee" title="Qabee - oromo" lang="om" hreflang="om" data-title="Qabee" data-language-autonym="Oromoo" data-language-local-name="oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%86%E0%A8%87%E0%A8%A4%E0%A8%A8" title="ਆਇਤਨ - punjabi" lang="pa" hreflang="pa" data-title="ਆਇਤਨ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Obj%C4%99to%C5%9B%C4%87" title="Objętość - polonés" lang="pl" hreflang="pl" data-title="Objętość" data-language-autonym="Polski" data-language-local-name="polonés" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%AD%D8%AC%D9%85" title="حجم - Western Punjabi" lang="pnb" hreflang="pnb" data-title="حجم" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Volume" title="Volume - portugués" lang="pt" hreflang="pt" data-title="Volume" data-language-autonym="Português" data-language-local-name="portugués" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/P%27ulin" title="P'ulin - quechua" lang="qu" hreflang="qu" data-title="P'ulin" data-language-autonym="Runa Simi" data-language-local-name="quechua" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Volum_(geometrie)" title="Volum (geometrie) - romanés" lang="ro" hreflang="ro" data-title="Volum (geometrie)" data-language-autonym="Română" data-language-local-name="romanés" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-rsk mw-list-item"><a href="https://rsk.wikipedia.org/wiki/%D0%92%D0%BE%D0%BB%D1%83%D0%BC%D0%B5%D0%BD" title="Волумен - Pannonian Rusyn" lang="rsk" hreflang="rsk" data-title="Волумен" data-language-autonym="Руски" data-language-local-name="Pannonian Rusyn" class="interlanguage-link-target"><span>Руски</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9E%D0%B1%D1%8A%D1%91%D0%BC" title="Объём - rus" lang="ru" hreflang="ru" data-title="Объём" data-language-autonym="Русский" data-language-local-name="rus" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sa mw-list-item"><a href="https://sa.wikipedia.org/wiki/%E0%A4%86%E0%A4%AF%E0%A4%A4%E0%A4%A8%E0%A4%AE%E0%A5%8D" title="आयतनम् - sanscrit" lang="sa" hreflang="sa" data-title="आयतनम्" data-language-autonym="संस्कृतम्" data-language-local-name="sanscrit" class="interlanguage-link-target"><span>संस्कृतम्</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Vulumi" title="Vulumi - sicilian" lang="scn" hreflang="scn" data-title="Vulumi" data-language-autonym="Sicilianu" data-language-local-name="sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Vollum" title="Vollum - escossés" lang="sco" hreflang="sco" data-title="Vollum" data-language-autonym="Scots" data-language-local-name="escossés" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sd mw-list-item"><a href="https://sd.wikipedia.org/wiki/%D8%AD%D8%AC%D9%85" title="حجم - sindi" lang="sd" hreflang="sd" data-title="حجم" data-language-autonym="سنڌي" data-language-local-name="sindi" class="interlanguage-link-target"><span>سنڌي</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Volumen" title="Volumen - serbocroat" lang="sh" hreflang="sh" data-title="Volumen" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbocroat" 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/Volume" title="Volume - Simple English" lang="en-simple" hreflang="en-simple" data-title="Volume" 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-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Objem_(matematika)" title="Objem (matematika) - eslovac" lang="sk" hreflang="sk" data-title="Objem (matematika)" data-language-autonym="Slovenčina" data-language-local-name="eslovac" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Prostornina" title="Prostornina - eslovèn" lang="sl" hreflang="sl" data-title="Prostornina" data-language-autonym="Slovenščina" data-language-local-name="eslovèn" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Mumhu" title="Mumhu - shona" lang="sn" hreflang="sn" data-title="Mumhu" data-language-autonym="ChiShona" data-language-local-name="shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Mug" title="Mug - somali" lang="so" hreflang="so" data-title="Mug" data-language-autonym="Soomaaliga" data-language-local-name="somali" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/V%C3%ABllimi" title="Vëllimi - albanés" lang="sq" hreflang="sq" data-title="Vëllimi" 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%97%D0%B0%D0%BF%D1%80%D0%B5%D0%BC%D0%B8%D0%BD%D0%B0" title="Запремина - serbi" lang="sr" hreflang="sr" data-title="Запремина" data-language-autonym="Српски / srpski" data-language-local-name="serbi" 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/Volym" title="Volym - suedés" lang="sv" hreflang="sv" data-title="Volym" data-language-autonym="Svenska" data-language-local-name="suedés" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Mjao" title="Mjao - swahili" lang="sw" hreflang="sw" data-title="Mjao" data-language-autonym="Kiswahili" data-language-local-name="swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/Uobjynto%C5%9B%C4%87" title="Uobjyntość - silesià" lang="szl" hreflang="szl" data-title="Uobjyntość" data-language-autonym="Ślůnski" data-language-local-name="silesià" class="interlanguage-link-target"><span>Ślůnski</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AE%A9_%E0%AE%85%E0%AE%B3%E0%AE%B5%E0%AF%81" title="கன அளவு - tamol" lang="ta" hreflang="ta" data-title="கன அளவு" data-language-autonym="தமிழ்" data-language-local-name="tamol" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%98%E0%B0%A8%E0%B0%AA%E0%B0%B0%E0%B0%BF%E0%B0%AE%E0%B0%BE%E0%B0%A3%E0%B0%AE%E0%B1%81" title="ఘనపరిమాణము - telogó" lang="te" hreflang="te" data-title="ఘనపరిమాణము" data-language-autonym="తెలుగు" data-language-local-name="telogó" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D2%B2%D0%B0%D2%B7%D0%BC" title="Ҳаҷм - tajic" lang="tg" hreflang="tg" data-title="Ҳаҷм" data-language-autonym="Тоҷикӣ" data-language-local-name="tajic" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%9B%E0%B8%A3%E0%B8%B4%E0%B8%A1%E0%B8%B2%E0%B8%95%E0%B8%A3" title="ปริมาตร - tai" lang="th" hreflang="th" data-title="ปริมาตร" data-language-autonym="ไทย" data-language-local-name="tai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tk mw-list-item"><a href="https://tk.wikipedia.org/wiki/G%C3%B6wr%C3%BCm_birlikleri" title="Göwrüm birlikleri - turcmèn" lang="tk" hreflang="tk" data-title="Göwrüm birlikleri" data-language-autonym="Türkmençe" data-language-local-name="turcmèn" class="interlanguage-link-target"><span>Türkmençe</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Volyum" title="Volyum - tagal" lang="tl" hreflang="tl" data-title="Volyum" data-language-autonym="Tagalog" data-language-local-name="tagal" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Hacim" title="Hacim - turc" lang="tr" hreflang="tr" data-title="Hacim" data-language-autonym="Türkçe" data-language-local-name="turc" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9E%D0%B1%27%D1%94%D0%BC" title="Об'єм - ucrainés" lang="uk" hreflang="uk" data-title="Об'єм" data-language-autonym="Українська" data-language-local-name="ucrainés" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%AD%D8%AC%D9%85" title="حجم - ordó" lang="ur" hreflang="ur" data-title="حجم" data-language-autonym="اردو" data-language-local-name="ordó" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Hajm" title="Hajm - ozbèc" lang="uz" hreflang="uz" data-title="Hajm" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="ozbèc" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Vo%C5%82ume" title="Vołume - vènet" lang="vec" hreflang="vec" data-title="Vołume" data-language-autonym="Vèneto" data-language-local-name="vènet" class="interlanguage-link-target"><span>Vèneto</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Th%E1%BB%83_t%C3%ADch" title="Thể tích - vietnamian" lang="vi" hreflang="vi" data-title="Thể tích" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamian" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Bolyum" title="Bolyum - waray" lang="war" hreflang="war" data-title="Bolyum" data-language-autonym="Winaray" data-language-local-name="waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E4%BD%93%E7%A7%AF" title="体积 - xinès wu" lang="wuu" hreflang="wuu" data-title="体积" data-language-autonym="吴语" data-language-local-name="xinès 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/%E4%BD%93%E7%A7%AF" title="体积 - chinés" lang="zh" hreflang="zh" data-title="体积" data-language-autonym="中文" data-language-local-name="chinés" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Th%C3%A9-chek" title="Thé-chek - xinès min del sud" lang="nan" hreflang="nan" data-title="Thé-chek" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="xinès min del sud" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%AB%94%E7%A9%8D" title="體積 - cantonés" lang="yue" hreflang="yue" data-title="體積" data-language-autonym="粵語" data-language-local-name="cantonés" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q39297#sitelinks-wikipedia" title="Modificar los ligams interlenga" class="wbc-editpage">Modificar los ligams</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Espacis de noms"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Volum" title="Veire l’article [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Discutir:Volum&action=edit&redlink=1" rel="discussion" class="new" title="Discussion a prepaus d'aquesta pagina (la pagina non existís pas) [t]" accesskey="t"><span>Discussion</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Canvia la variant de llengua" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">occitan</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Afichatges"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Volum"><span>Legir</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Volum&veaction=edit" title="Modificar aquela pagina [v]" accesskey="v"><span>Modificar</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Volum&action=edit" title="Modificar lo còdi font d'aquela pagina [e]" accesskey="e"><span>Modificar lo còdi</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Volum&action=history" title="Los autors e versions precedentas d'aquesta pagina. [h]" accesskey="h"><span>Veire l'istoric</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Eines de la pàgina"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Bóstia d'aisinas" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Bóstia d'aisinas</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Aisinas</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">mòu a la barra laterala</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">Escondre</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Mai d'opcions" > <div class="vector-menu-heading"> Accions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Volum"><span>Legir</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Volum&veaction=edit" title="Modificar aquela pagina [v]" accesskey="v"><span>Modificar</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Volum&action=edit" title="Modificar lo còdi font d'aquela pagina [e]" accesskey="e"><span>Modificar lo còdi</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Volum&action=history"><span>Veire l'istoric</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General - Generau </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Especial:Paginas_ligadas/Volum" title="Lista de las paginas ligadas a aquesta. [j]" accesskey="j"><span>Paginas connèxas</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Especial:Seguit_dels_ligams/Volum" rel="nofollow" title="Lista dels darrièrs cambiaments de las paginas ligadas a aquesta [k]" accesskey="k"><span>Seguiment dels ligams</span></a></li><li id="t-upload" class="mw-list-item"><a href="//commons.wikimedia.org/wiki/Special:UploadWizard?uselang=oc" title="Mandar un imatge o fichièr mèdia sul servidor [u]" accesskey="u"><span>Importar un fichièr</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Volum&oldid=2442366" title="Ligam permanent cap a aquesta version de la pagina"><span>Ligam permanent</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Volum&action=info" title="Mai d’informacion sus aquesta pagina"><span>Informacion sus la pagina</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Especial:Citar&page=Volum&id=2442366&wpFormIdentifier=titleform" title="Informacions sus cossí citar aquesta pagina"><span>Citar aqueste article</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Especial:UrlShortener&url=https%3A%2F%2Foc.wikipedia.org%2Fwiki%2FVolum"><span>Obténer una URL acorchida</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Especial:QrCode&url=https%3A%2F%2Foc.wikipedia.org%2Fwiki%2FVolum"><span>Telecargar lo còdi QR</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Imprimir / exportar </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Especial:Libre&bookcmd=book_creator&referer=Volum"><span>Crear un libre</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Especial:DownloadAsPdf&page=Volum&action=show-download-screen"><span>Telecargar coma PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Volum&printable=yes" title="Version imprimibla d'aquesta pagina [p]" accesskey="p"><span>Imprimir</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> Dins d'autres projèctes </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Volumes" hreflang="en"><span>Wikimedia Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q39297" title="Ligam cap a l’element de referencial de las donadas connectadas [g]" accesskey="g"><span>Element Wikidata</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Eines de la pàgina"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Aparéncia"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Aparéncia</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">mòu a la barra laterala</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">Escondre</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">Un article de Wikipèdia, l'enciclopèdia liura.</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="oc" dir="ltr"><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichi%C3%A8r:Simple_Measuring_Cup.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/27/Simple_Measuring_Cup.jpg/220px-Simple_Measuring_Cup.jpg" decoding="async" width="220" height="255" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/27/Simple_Measuring_Cup.jpg/330px-Simple_Measuring_Cup.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/27/Simple_Measuring_Cup.jpg/440px-Simple_Measuring_Cup.jpg 2x" data-file-width="1020" data-file-height="1180" /></a><figcaption>Un vas que se pòt utilizar per mesurar de volums de <a href="/w/index.php?title=L%C3%ADquid&action=edit&redlink=1" class="new" title="Líquid (la pagina existís pas)">líquids</a></figcaption></figure> <p>Lo <b>volum</b> (var. <b>volume</b>) es la porcion o quantitat d'<a href="/w/index.php?title=Espaci_tridimensional&action=edit&redlink=1" class="new" title="Espaci tridimensional (la pagina existís pas)">espaci tridimensional</a> tancat dins una frontièra. Per exemple, lo volum es l'<a href="/wiki/Espaci" title="Espaci">espaci</a> o forma qu'una substància (<a href="/wiki/Solid" title="Solid">solid</a>, <a href="/wiki/Liquid" title="Liquid">liquid</a>, <a href="/wiki/Gas" title="Gas">gas</a> o <a href="/wiki/Plasma_(fisica)" title="Plasma (fisica)">plasma</a>) ocupa o conten.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Lo volum es abitualament quantificat numericament utilizant l'unitat derivada del <a href="/wiki/Sist%C3%A8ma_Internacional_d%27Unitats" class="mw-redirect" title="Sistèma Internacional d'Unitats">Sistèma Internacional d'Unitats</a>, lo <a href="/wiki/M%C3%A8tre_cubic" title="Mètre cubic">mètre cubic</a>. </p><p>Las formas matematicas plaçadas dins l'<a href="/wiki/Espaci" title="Espaci">espaci</a> donan luòc a de volums. Los volums formats per de figuras simplas –com las formas regularas, circularas o d'arèstas dechas– se pòdon calcular aisidament utilisant de formulas matematicas. D'autra costat, las formas mai complicadas se tròban mejans lo <a href="/w/index.php?title=Calcul_integral&action=edit&redlink=1" class="new" title="Calcul integral (la pagina existís pas)">calcul integral</a> s'existís una formula per la frontièra. Las figuras d'una dimension (coma la <a href="/w/index.php?title=Linha_(matemeticas)&action=edit&redlink=1" class="new" title="Linha (matemeticas) (la pagina existís pas)">linha</a>) e las formas de doas dimensions (coma los <a href="/w/index.php?title=Carrat_(matematicas)&action=edit&redlink=1" class="new" title="Carrat (matematicas) (la pagina existís pas)">carrats</a>) an un volum zèro dins l'espaci tridimensional. </p><p>Lo volum d'un solid (que siá de forma regulara o irregulara) se pòt determinar a partir del <a href="/w/index.php?title=Despla%C3%A7ament_(fluid)&action=edit&redlink=1" class="new" title="Desplaçament (fluid) (la pagina existís pas)">desplaçament de fluid</a>. Lo desplaçament de liquid tanben se pòt utilizar per determinar lo volum d'un gas. Lo volum combinat de doas substàncias es generalament mai elevat que lo volum d'una de las substàncias. Pasmens, a vegadas una substància se dissòlv dins l'autra, atal lo volum, dins aquel cas, es pas <a href="/w/index.php?title=Funcion_additiva&action=edit&redlink=1" class="new" title="Funcion additiva (la pagina existís pas)">additiu</a>. </p><p>En <a href="/w/index.php?title=Geometria_diferenciala&action=edit&redlink=1" class="new" title="Geometria diferenciala (la pagina existís pas)">geometria diferenciala</a>, lo volum s'exprima dins de tèrmes de <a href="/w/index.php?title=Forma_volum&action=edit&redlink=1" class="new" title="Forma volum (la pagina existís pas)">forma volum</a>, e es un <a href="/w/index.php?title=Invariant_(matemeticas)&action=edit&redlink=1" class="new" title="Invariant (matemeticas) (la pagina existís pas)">invariant</a> <a href="/w/index.php?title=Geometria_de_Riemann&action=edit&redlink=1" class="new" title="Geometria de Riemann (la pagina existís pas)">riemanniana</a> globala importanta. D'autra costat, en <a href="/wiki/Termodinamica" title="Termodinamica">termodinamica</a>, lo volum es un <a href="/w/index.php?title=Volum_de_gas&action=edit&redlink=1" class="new" title="Volum de gas (la pagina existís pas)">paremètre fonamental</a>, e es la <a href="/w/index.php?title=Variabla_conjugada&action=edit&redlink=1" class="new" title="Variabla conjugada (la pagina existís pas)">variabla conjugada</a> de la <a href="/wiki/Pression" title="Pression">pression</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Unitats">Unitats</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Volum&veaction=edit&section=1" title="Modificar la seccion : Unitats" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Volum&action=edit&section=1" title="Editar lo còdi de la seccion seguenta: Unitats"><span>modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Quina que siá l'<a href="/w/index.php?title=Unitat_de_longor&action=edit&redlink=1" class="new" title="Unitat de longor (la pagina existís pas)">unitat de longor</a> a la seuna correspondéncia en <a href="/w/index.php?title=Unitat_de_volum&action=edit&redlink=1" class="new" title="Unitat de volum (la pagina existís pas)">unitat de volum</a>, normalament lo volum del <a href="/wiki/Cube" title="Cube">cube</a> l'arèsta qu'a la longor donada. Per exemple, un <a href="/w/index.php?title=Centim%C3%A8tre_cubic&action=edit&redlink=1" class="new" title="Centimètre cubic (la pagina existís pas)">centimètre cubic</a> (cm³) es lo volum del cube que las arestas mesuran 1 <a href="/w/index.php?title=Centimetre&action=edit&redlink=1" class="new" title="Centimetre (la pagina existís pas)">cm</a> de longor. </p><p>Dins lo <a href="/wiki/Sist%C3%A8ma_Internacional_d%27Unitats" class="mw-redirect" title="Sistèma Internacional d'Unitats">Sistèma Internacional d'Unitats</a> (SI), l'unitat estandard de volum es lo mèstre cubic (m³). Lo <a href="/wiki/Sist%C3%A8ma_metric" class="mw-redirect" title="Sistèma metric">sistèma metric</a> tanben inclutz lo <a href="/wiki/Litre" title="Litre">litre</a> (l) coma unitat de volum; un litre es lo volum d'un cube d'arèsta detz centimetres (un <a href="/w/index.php?title=Decimetre_cubic&action=edit&redlink=1" class="new" title="Decimetre cubic (la pagina existís pas)">decimetre cubic</a>). Alara: </p> <dl><dd>1 litre = (10 cm)³ = 1000 centimetres cubics = 0.001 metres cubics,</dd></dl> <p>llavors </p> <dl><dd>1 metre cubic = 1000 litres</dd></dl> <p>Las quantitats pichonas de liquid se meruran en <a href="/w/index.php?title=Millilitres&action=edit&redlink=1" class="new" title="Millilitres (la pagina existís pas)">millilitres</a>, onte </p> <dl><dd>1 millilitre = 0,001 litres = 1 centimetre cubic</dd></dl> <p>D'autres unitats tradicionalas de volum dins d'autres sistèmas qu'encara son en vigors dins qualques païses son entre autre las seguentas: <a href="/w/index.php?title=Poce_cubic&action=edit&redlink=1" class="new" title="Poce cubic (la pagina existís pas)">poce cubic</a>, <a href="/w/index.php?title=P%C3%A8_cubic&action=edit&redlink=1" class="new" title="Pè cubic (la pagina existís pas)">pè cubic</a>, <a href="/w/index.php?title=On%C3%A7a_liquida&action=edit&redlink=1" class="new" title="Onça liquida (la pagina existís pas)">onça liquida</a>, <a href="/w/index.php?title=Pinta_(unitat_de_volum)&action=edit&redlink=1" class="new" title="Pinta (unitat de volum) (la pagina existís pas)">pinta</a>, <a href="/w/index.php?title=Quart_(unitat)&action=edit&redlink=1" class="new" title="Quart (unitat) (la pagina existís pas)">quart</a>, <a href="/w/index.php?title=Galon_(unitat)&action=edit&redlink=1" class="new" title="Galon (unitat) (la pagina existís pas)">galon</a>], <a href="/w/index.php?title=Barril_(unitat)&action=edit&redlink=1" class="new" title="Barril (unitat) (la pagina existís pas)">barril</a>, <a href="/w/index.php?title=C%C3%B2rda_(unitat)&action=edit&redlink=1" class="new" title="Còrda (unitat) (la pagina existís pas)">còrda</a>, <a href="/w/index.php?title=Peck&action=edit&redlink=1" class="new" title="Peck (la pagina existís pas)">peck</a>, <a href="/w/index.php?title=Bushel&action=edit&redlink=1" class="new" title="Bushel (la pagina existís pas)">bushel</a> o <a href="/w/index.php?title=Hogshead&action=edit&redlink=1" class="new" title="Hogshead (la pagina existís pas)">hogshead</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Fórmulas_per_calcular_de_volums"><span id="F.C3.B3rmulas_per_calcular_de_volums"></span>Fórmulas per calcular de volums</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Volum&veaction=edit&section=2" title="Modificar la seccion : Fórmulas per calcular de volums" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Volum&action=edit&section=2" title="Editar lo còdi de la seccion seguenta: Fórmulas per calcular de volums"><span>modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable"> <tbody><tr> <th>Cos</th> <th>Formula del volum</th> <th>Variables </th></tr> <tr> <td><a href="/wiki/Cube" title="Cube">Cube</a> </td> <td style="text-align:center"><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^{3}\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mspace width="thickmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{3}\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f80731f16bb951a9dae465d1def439c3f8e1f89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.929ex; height:2.676ex;" alt="{\displaystyle a^{3}\;}" /></span> </td> <td><i>a</i> = longor d'una arèsta </td></tr> <tr> <td><a href="/wiki/Cilindre_(geometria)" title="Cilindre (geometria)">Cilindre</a> </td> <td style="text-align:center"><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 \pi r^{2}h\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>h</mi> <mspace width="thickmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi r^{2}h\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6577ed24266c4e899663056b68646265170e0c4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.419ex; height:2.676ex;" alt="{\displaystyle \pi r^{2}h\;}" /></span> </td> <td><i>r</i> = radi de la fàcia circulara, <i>h</i> = nautor </td></tr> <tr> <td><a href="/w/index.php?title=Prisme_(geometria)&action=edit&redlink=1" class="new" title="Prisme (geometria) (la pagina existís pas)">Prisme</a> </td> <td style="text-align:center"><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\cdot h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>⋅</mo> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\cdot h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8546ac33d5cc63f312dc46e7f272cb8a7ff9163f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.782ex; height:2.176ex;" alt="{\displaystyle B\cdot h}" /></span> </td> <td><i>B</i> = aira de la basa, <i>h</i> = nautor </td></tr> <tr> <td><a href="/w/index.php?title=Prisme_rectangular&action=edit&redlink=1" class="new" title="Prisme rectangular (la pagina existís pas)">Prisme rectangular</a> </td> <td style="text-align:center"><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 l\cdot w\cdot h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mo>⋅</mo> <mi>w</mi> <mo>⋅</mo> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l\cdot w\cdot h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/665d8994ec0ab3a941e4a67728f26fd4e2897bbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.055ex; height:2.176ex;" alt="{\displaystyle l\cdot w\cdot h}" /></span> </td> <td>l = longor, w = largor, h = nautor </td></tr> <tr> <td><a href="/wiki/Esf%C3%A8ra" title="Esfèra">Esfèra</a> </td> <td style="text-align:center"><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 {\frac {4}{3}}\pi r^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {4}{3}}\pi r^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7112cb504e1399319cb9edbc622d1c4609fd56e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.433ex; height:5.176ex;" alt="{\displaystyle {\frac {4}{3}}\pi r^{3}}" /></span> </td> <td><i>r</i> = radi de l'esfèra<br />qu'es l'<a href="/w/index.php?title=Integrala&action=edit&redlink=1" class="new" title="Integrala (la pagina existís pas)">integrala</a> de l'<a href="/w/index.php?title=Aira_superficiala&action=edit&redlink=1" class="new" title="Aira superficiala (la pagina existís pas)">aira superficiala</a> de l'esfèra </td></tr> <tr> <td><a href="/w/index.php?title=Ellipso%C3%ADd&action=edit&redlink=1" class="new" title="Ellipsoíd (la pagina existís pas)">Ellipsoíd</a> </td> <td style="text-align:center"><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 {\frac {4}{3}}\pi abc}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mi>π</mi> <mi>a</mi> <mi>b</mi> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {4}{3}}\pi abc}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6bab0d7b6f5d52c794e589223f04011463277e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.565ex; height:5.176ex;" alt="{\displaystyle {\frac {4}{3}}\pi abc}" /></span> </td> <td><i>a</i>, <i>b</i>, <i>c</i> = semiaxes de l'ellipsoíd </td></tr> <tr> <td><a href="/wiki/Piramida" title="Piramida">Piramida</a> </td> <td style="text-align:center"><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 {\frac {1}{3}}Bh}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mi>B</mi> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{3}}Bh}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31fdab0536eaf11c26391cac9ae01644f380e9ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.102ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{3}}Bh}" /></span> </td> <td><i>B</i> = aira de la basa, <i>h</i> = nautor de la piramida </td></tr> <tr> <td><a href="/wiki/C%C3%B2n" title="Còn">Còn</a> </td> <td style="text-align:center"><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 {\frac {1}{3}}\pi r^{2}h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{3}}\pi r^{2}h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d2006a2f5533ecd0dce30332eba227da86b7404" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.772ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{3}}\pi r^{2}h}" /></span> </td> <td><i>r</i> = radi del cercle de la basa, <i>h</i> = distància de la base al vertèx (nautor) </td></tr> <tr> <td><a href="/wiki/Tetra%C3%A8dre" title="Tetraèdre">Tetraèdre</a><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </td> <td style="text-align:center"><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 {2}} \over 12}a^{3}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>12</mn> </mfrac> </mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {{\sqrt {2}} \over 12}a^{3}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4981067eb214e485499952bda8fed98775873f94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.606ex; height:5.843ex;" alt="{\displaystyle {{\sqrt {2}} \over 12}a^{3}\,}" /></span> </td> <td><i>a</i> = longor de l'arèsta </td></tr> <tr> <td><a href="/w/index.php?title=Parallelepep%C3%A8d&action=edit&redlink=1" class="new" title="Parallelepepèd (la pagina existís pas)">Parallelepepèd</a> </td> <td style="text-align:center"><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 abc{\sqrt {K}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>K</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle abc{\sqrt {K}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3e4b8b3b1cde51f83356ccc7a33c0ae01c4445d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.236ex; height:3.009ex;" alt="{\displaystyle abc{\sqrt {K}}}" /></span> <p><br /> <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 {\begin{aligned}K=&1+2\cos(\alpha )\cos(\beta )\cos(\gamma )\\&-\cos ^{2}(\alpha )-\cos ^{2}(\beta )-\cos ^{2}(\gamma )\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>K</mi> <mo>=</mo> </mtd> <mtd> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>−</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}K=&1+2\cos(\alpha )\cos(\beta )\cos(\gamma )\\&-\cos ^{2}(\alpha )-\cos ^{2}(\beta )-\cos ^{2}(\gamma )\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ac7ae70124e4ef3c0338247e5d92c171280bd10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.798ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}K=&1+2\cos(\alpha )\cos(\beta )\cos(\gamma )\\&-\cos ^{2}(\alpha )-\cos ^{2}(\beta )-\cos ^{2}(\gamma )\end{aligned}}}" /></span> </p> </td> <td><i>a</i>, <i>b</i> e <i>c</i> son las longors de las arèstas, e α, β e γ los angles intèrnes entre eles<br /><i>Nòta: tenent los <a href="/wiki/Vector_(matematicas)" title="Vector (matematicas)">vectors</a> directors non coplanars de las tres arèstas <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 ({\vec {v}}_{1},{\vec {v}}_{2},{\vec {v}}_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\vec {v}}_{1},{\vec {v}}_{2},{\vec {v}}_{3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bc83356af91a36ab749f5cca95df6c1c905aa0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.566ex; height:2.843ex;" alt="{\displaystyle ({\vec {v}}_{1},{\vec {v}}_{2},{\vec {v}}_{3})}" /></span>, lo volum se pòt calcular a partir del <a href="/w/index.php?title=Produch_mixte&action=edit&redlink=1" class="new" title="Produch mixte (la pagina existís pas)">produch mixte</a> dels tres vectors: <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 V=|\det({\vec {v}}_{1},{\vec {v}}_{2},{\vec {v}}_{3})|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=|\det({\vec {v}}_{1},{\vec {v}}_{2},{\vec {v}}_{3})|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea287779a62fd00eddad12f84587766bc54ba75c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.362ex; height:2.843ex;" alt="{\displaystyle V=|\det({\vec {v}}_{1},{\vec {v}}_{2},{\vec {v}}_{3})|}" /></span>.</i> </td></tr> <tr> <td>Quina que siá figura generada per limpament<br />(cal <a href="/w/index.php?title=Calcul_integral&action=edit&redlink=1" class="new" title="Calcul integral (la pagina existís pas)">calcul integral</a>) </td> <td style="text-align:center"><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 \int _{a}^{b}A(h)\,\mathrm {d} h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>A</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}A(h)\,\mathrm {d} h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf389e165e67594607a07e96126d3b8d16da0067" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.699ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}A(h)\,\mathrm {d} h}" /></span> </td> <td><i>h</i> = quina que siá dimension de la figura,<br /><i>A</i>(<i>h</i>) = aira de las seccions transversalas perpendicularas a <i>h</i> descrichas coma una foncion lo long d'<i>h</i>. <i>a</i> e <i>b</i> son las limitas d'integracion per lo limpament volumetric.<br />(Aquò foncionarà per quina que siá figura se la seuna aira transversala se pòt determinar dempuèi d'<i>h</i>). </td></tr> <tr> <td>Quina que siá figura generada per rotacion<br />(cal <a href="/w/index.php?title=Calcul_integral&action=edit&redlink=1" class="new" title="Calcul integral (la pagina existís pas)">calcul integral</a>) </td> <td><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 \pi \int _{a}^{b}\left({\left[R_{O}(x)\right]}^{2}-{\left[R_{I}(x)\right]}^{2}\right)\mathrm {d} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \int _{a}^{b}\left({\left[R_{O}(x)\right]}^{2}-{\left[R_{I}(x)\right]}^{2}\right)\mathrm {d} x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15f622970c61715e48e9c33088619c89a4bd40e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.182ex; height:6.343ex;" alt="{\displaystyle \pi \int _{a}^{b}\left({\left[R_{O}(x)\right]}^{2}-{\left[R_{I}(x)\right]}^{2}\right)\mathrm {d} x}" /></span> </td> <td><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_{O}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{O}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a131c871300f6551262a598d82b3847a8ead4e50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.25ex; height:2.509ex;" alt="{\displaystyle R_{O}}" /></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 R_{I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f9ea449880cf4ea2cad83bcdb8ecf27ded1b2b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.825ex; height:2.509ex;" alt="{\displaystyle R_{I}}" /></span> son las foncions qu'exprimisson los radis extèrne e intèrne de la foncion, respectivament. </td></tr> <tr> <td><a href="/w/index.php?title=Ampola_de_Klein&action=edit&redlink=1" class="new" title="Ampola de Klein (la pagina existís pas)">Ampola de Klein</a> </td> <td style="text-align:center"><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 0\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mspace width="thickmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9373a19fcb6697cfcf6717983f428b5b95bcb078" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle 0\;}" /></span> </td> <td>A pas volum (a pas d'interior) </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Volum_dels_solids_platonics">Volum dels solids platonics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Volum&veaction=edit&section=3" title="Modificar la seccion : Volum dels solids platonics" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Volum&action=edit&section=3" title="Editar lo còdi de la seccion seguenta: Volum dels solids platonics"><span>modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Los <a href="/wiki/Solid_platonic" title="Solid platonic">solids platonics</a> comprenon los cinc unics polièdres regulars. Se l'arèsta del polièdre es <i>a</i>, lo seu volum se dona per la taula seguenta: </p> <table class="wikitable"> <tbody><tr> <th>Polièdre </th> <th>Volum </th> <th>Imatge </th> <th>Polièdre </th> <th>Volum </th> <th>Imatge </th></tr> <tr> <td width="100px"><a href="/wiki/Tetra%C3%A8dre" title="Tetraèdre">Tetraèdre</a> </td> <td width="200px"><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 V={\frac {1}{12}}{\sqrt {2}}a^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>12</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\frac {1}{12}}{\sqrt {2}}a^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1141d43793b2e407441692b18f8e5b155bfa27f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.429ex; height:5.176ex;" alt="{\displaystyle V={\frac {1}{12}}{\sqrt {2}}a^{3}}" /></span> </td> <td width="120px"><figure class="mw-default-size mw-halign-right" typeof="mw:File"><a href="/wiki/Fichi%C3%A8r:120px-Tetrahedron-slowturn.gif" class="mw-file-description" title="Tetraèdre"><img alt="Tetraèdre" src="//upload.wikimedia.org/wikipedia/commons/9/9e/120px-Tetrahedron-slowturn.gif" decoding="async" width="120" height="120" class="mw-file-element" data-file-width="120" data-file-height="120" /></a><figcaption>Tetraèdre</figcaption></figure> </td> <td width="100px"><a href="/w/index.php?title=Dodeca%C3%A8dre_regular&action=edit&redlink=1" class="new" title="Dodecaèdre regular (la pagina existís pas)">Dodecaèdre regular</a> </td> <td width="200px"><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 V={\frac {1}{4}}(15+7{\sqrt {5}})a^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>15</mn> <mo>+</mo> <mn>7</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\frac {1}{4}}(15+7{\sqrt {5}})a^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0c5e60311cc0e52a79b7164700947d88f00056f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.404ex; height:5.176ex;" alt="{\displaystyle V={\frac {1}{4}}(15+7{\sqrt {5}})a^{3}}" /></span> </td> <td width="120px"><figure class="mw-default-size mw-halign-right" typeof="mw:File"><a href="/wiki/Fichi%C3%A8r:120px-Dodecahedron-slowturn.gif" class="mw-file-description" title="Dodecaèdre regular"><img alt="Dodecaèdre regular" src="//upload.wikimedia.org/wikipedia/commons/7/70/120px-Dodecahedron-slowturn.gif" decoding="async" width="120" height="120" class="mw-file-element" data-file-width="120" data-file-height="120" /></a><figcaption>Dodecaèdre regular</figcaption></figure> </td></tr> <tr> <td><a href="/wiki/Cube" title="Cube">Cube</a> </td> <td><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 V=a^{3}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=a^{3}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8190db0d682a966215231c1441b4a7fb7f795f2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.557ex; height:2.676ex;" alt="{\displaystyle V=a^{3}\,}" /></span> </td> <td><figure class="mw-default-size mw-halign-right" typeof="mw:File"><a href="/wiki/Fichi%C3%A8r:120px-Hexahedron-slowturn.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/50/120px-Hexahedron-slowturn.gif" decoding="async" width="120" height="120" class="mw-file-element" data-file-width="120" data-file-height="120" /></a><figcaption></figcaption></figure> </td> <td><a href="/w/index.php?title=Icosa%C3%A8dre&action=edit&redlink=1" class="new" title="Icosaèdre (la pagina existís pas)">Icosaèdre</a> </td> <td><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 V={\frac {5}{6}}\varphi ^{2}a^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>6</mn> </mfrac> </mrow> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\frac {5}{6}}\varphi ^{2}a^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4895e839164030b14d173b15698b10636956221f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.743ex; height:5.176ex;" alt="{\displaystyle V={\frac {5}{6}}\varphi ^{2}a^{3}}" /></span> on <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 \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" /></span> es lo <a href="/wiki/Nombre_d%27aur" title="Nombre d'aur">nombre d'aur</a> </td> <td><figure class="mw-default-size mw-halign-right" typeof="mw:File"><a href="/wiki/Fichi%C3%A8r:120px-Icosahedron-slowturn.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/7/73/120px-Icosahedron-slowturn.gif" decoding="async" width="120" height="120" class="mw-file-element" data-file-width="120" data-file-height="120" /></a><figcaption></figcaption></figure> </td></tr> <tr> <td><a href="/w/index.php?title=Octa%C3%A8dre&action=edit&redlink=1" class="new" title="Octaèdre (la pagina existís pas)">Octaèdre</a> </td> <td><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 V={\frac {1}{3}}{\sqrt {2}}a^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\frac {1}{3}}{\sqrt {2}}a^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e865c377b785630cf620ab8fceba14ac3807e77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.267ex; height:5.176ex;" alt="{\displaystyle V={\frac {1}{3}}{\sqrt {2}}a^{3}}" /></span> </td> <td><figure class="mw-default-size mw-halign-right" typeof="mw:File"><a href="/wiki/Fichi%C3%A8r:120px-Octahedron-slowturn.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/4/41/120px-Octahedron-slowturn.gif" decoding="async" width="120" height="120" class="mw-file-element" data-file-width="120" data-file-height="120" /></a><figcaption></figcaption></figure> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Proporcion_entre_los_volums_d'un_còn,_esfèra_e_cilindre_del_meteis_radi_e_nautor"><span id="Proporcion_entre_los_volums_d.27un_c.C3.B2n.2C_esf.C3.A8ra_e_cilindre_del_meteis_radi_e_nautor"></span>Proporcion entre los volums d'un còn, esfèra e cilindre del meteis radi e nautor</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Volum&veaction=edit&section=4" title="Modificar la seccion : Proporcion entre los volums d'un còn, esfèra e cilindre del meteis radi e nautor" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Volum&action=edit&section=4" title="Editar lo còdi de la seccion seguenta: Proporcion entre los volums d'un còn, esfèra e cilindre del meteis radi e nautor"><span>modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichi%C3%A8r:Inscribed_cone_sphere_cylinder.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Inscribed_cone_sphere_cylinder.svg/330px-Inscribed_cone_sphere_cylinder.svg.png" decoding="async" width="330" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Inscribed_cone_sphere_cylinder.svg/495px-Inscribed_cone_sphere_cylinder.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/12/Inscribed_cone_sphere_cylinder.svg/660px-Inscribed_cone_sphere_cylinder.svg.png 2x" data-file-width="512" data-file-height="256" /></a><figcaption>Un còn, esfèra e cilindre de radi <i>r</i> e nautor <i>h</i></figcaption></figure> <p>Las formulas anterioras se pòdon utilizar per demostrar que los volums d'un <a href="/wiki/C%C3%B2n" title="Còn">còn</a>, <a href="/wiki/Esf%C3%A8ra" title="Esfèra">esfèra</a> e <a href="/wiki/Cilindre_(geometria)" title="Cilindre (geometria)">cilindre</a> del meteis radi e nautor seguisson la proporcion 1 : 2 : 3. La demostracion es la seguenta: siá lo radi <i>r</i> e la nautor <i>h</i> (que per l'esfèra es 2<i>r</i>). Lo volum del còn es: </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 {\tfrac {1}{3}}\pi r^{2}h={\tfrac {1}{3}}\pi r^{2}(2r)=({\tfrac {2}{3}}\pi r^{3})\times 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>h</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>2</mn> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>×</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{3}}\pi r^{2}h={\tfrac {1}{3}}\pi r^{2}(2r)=({\tfrac {2}{3}}\pi r^{3})\times 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c74fb7dd4eba1bf760bb56bf70ea30a536a9b1bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:32.647ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{3}}\pi r^{2}h={\tfrac {1}{3}}\pi r^{2}(2r)=({\tfrac {2}{3}}\pi r^{3})\times 1}" /></span></dd></dl> <p>Lo volum de l'esfèra es: </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 {\tfrac {4}{3}}\pi r^{3}=({\tfrac {2}{3}}\pi r^{3})\times 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>×</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {4}{3}}\pi r^{3}=({\tfrac {2}{3}}\pi r^{3})\times 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b0e7a747f2133f65a86f5bca249449feeef32ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:19.097ex; height:3.676ex;" alt="{\displaystyle {\tfrac {4}{3}}\pi r^{3}=({\tfrac {2}{3}}\pi r^{3})\times 2}" /></span></dd></dl> <p>Lo volum del cilindre es: </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 \pi r^{2}h=\pi r^{2}(2r)=({\tfrac {2}{3}}\pi r^{3})\times 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>h</mi> <mo>=</mo> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>2</mn> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>×</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi r^{2}h=\pi r^{2}(2r)=({\tfrac {2}{3}}\pi r^{3})\times 3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8724568655365575441cb7bc465501a5cc2334fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:29.331ex; height:3.676ex;" alt="{\displaystyle \pi r^{2}h=\pi r^{2}(2r)=({\tfrac {2}{3}}\pi r^{3})\times 3}" /></span></dd></dl> <p>La descobèrta de la proporcion 2 : 3 entre los volums de l'esfèra e lo cilindre s'atribuís a <a href="/wiki/Arquim%C3%A8des" title="Arquimèdes">Arquimèdes</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Calcul_del_volum_per_integralas">Calcul del volum per integralas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Volum&veaction=edit&section=5" title="Modificar la seccion : Calcul del volum per integralas" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Volum&action=edit&section=5" title="Editar lo còdi de la seccion seguenta: Calcul del volum per integralas"><span>modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>L'integrala multiple es una partida limitada 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 \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}" /></span>, lo volum del cilindre qu'a per generatriu la frontièra 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 {\mathcal {D}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3277962e1959c3241fb1b70c7f0ac6dcefebd966" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.792ex; height:2.176ex;" alt="{\displaystyle {\mathcal {D}}}" /></span>, delimitat pel plan <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 z=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b92bfc06485cc90286474b14a516a68d8bfdd7b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.349ex; height:2.176ex;" alt="{\displaystyle z=0}" /></span> e la superfícia d'equacion <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 z=f(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=f(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1eefb2840000f404c8c0f3f5d6d72f2624854591" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.794ex; height:2.843ex;" alt="{\displaystyle z=f(x,y)}" /></span> –amb <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" /></span> positiva e continua sus <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 {\mathcal {D}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3277962e1959c3241fb1b70c7f0ac6dcefebd966" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.792ex; height:2.176ex;" alt="{\displaystyle {\mathcal {D}}}" /></span>– es: </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 V=\iint _{\mathcal {D}}f(x,y)\,\mathrm {d} x\,\mathrm {d} y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <msub> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=\iint _{\mathcal {D}}f(x,y)\,\mathrm {d} x\,\mathrm {d} y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c35dec4a1848f873ee903f79f73c355b9b0a2c7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.742ex; height:5.676ex;" alt="{\displaystyle V=\iint _{\mathcal {D}}f(x,y)\,\mathrm {d} x\,\mathrm {d} y}" /></span></dd></dl> <p>Dins lo cas que lo domèni <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 {\mathcal {D}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3277962e1959c3241fb1b70c7f0ac6dcefebd966" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.792ex; height:2.176ex;" alt="{\displaystyle {\mathcal {D}}}" /></span> es definit per las condicions simplas <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}<x<x_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <mi>x</mi> <mo><</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}<x<x_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a1322657cfe23659f1369f22472e939fd5d7a4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.294ex; height:2.176ex;" alt="{\displaystyle x_{1}<x<x_{2}}" /></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 y_{1}(x)<y(x)<y_{2}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi>y</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo><</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{1}(x)<y(x)<y_{2}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/618e2fd8ab24983b858297aa070b38cb38b9f1b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.156ex; height:2.843ex;" alt="{\displaystyle y_{1}(x)<y(x)<y_{2}(x)}" /></span>, lo calcul se reduch a: </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 V=\int _{x_{1}}^{x_{2}}\!\int _{y_{1}(x)}^{y_{2}(x)}f(x,y)\,\mathrm {d} y\,\mathrm {d} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mspace width="negativethinmathspace"></mspace> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=\int _{x_{1}}^{x_{2}}\!\int _{y_{1}(x)}^{y_{2}(x)}f(x,y)\,\mathrm {d} y\,\mathrm {d} x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/787b391911c770088a60a6f45092333512237c27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.746ex; height:6.676ex;" alt="{\displaystyle V=\int _{x_{1}}^{x_{2}}\!\int _{y_{1}(x)}^{y_{2}(x)}f(x,y)\,\mathrm {d} y\,\mathrm {d} x}" /></span></dd></dl> <p>Se <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 {\mathcal {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}" /></span> es una partida limitada 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 \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}" /></span> e la foncion constanta 1 es integrable sus <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 {\mathcal {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}" /></span>, lo volum 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 {\mathcal {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}" /></span> es alara: </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 V=\iiint _{\mathcal {A}}\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <msub> <mo>∭</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=\iiint _{\mathcal {A}}\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3ec74526d5ca911cf6f224af522734e4b04cc07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.775ex; height:5.843ex;" alt="{\displaystyle V=\iiint _{\mathcal {A}}\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z}" /></span></dd></dl> <p>Dins lo cas que lo domèni <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 {\mathcal {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}" /></span> es definit per las condicions simples <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}(z,y)<x(z,y)<x_{2}(z,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo><</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}(z,y)<x(z,y)<x_{2}(z,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8df131e97b8f53d2919e6be92fcb948deef22e18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.555ex; height:2.843ex;" alt="{\displaystyle x_{1}(z,y)<x(z,y)<x_{2}(z,y)}" /></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 y_{1}(z)<y(z)<y_{2}(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi>y</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo><</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{1}(z)<y(z)<y_{2}(z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f22cafabb233cfbd23a2ee269f0c88aaaa544bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.432ex; height:2.843ex;" alt="{\displaystyle y_{1}(z)<y(z)<y_{2}(z)}" /></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 z_{1}<z<z_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <mi>z</mi> <mo><</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{1}<z<z_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1702f3066379892139180e29afe157debee6350e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.556ex; height:2.176ex;" alt="{\displaystyle z_{1}<z<z_{2}}" /></span>, aquel calcul se redutz a: </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 V=\int _{z_{1}}^{z_{2}}\!\int _{y_{1}(z)}^{y_{2}(z)}\!\int _{x_{1}(z,y)}^{x_{2}(z,y)}\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mspace width="negativethinmathspace"></mspace> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mspace width="negativethinmathspace"></mspace> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=\int _{z_{1}}^{z_{2}}\!\int _{y_{1}(z)}^{y_{2}(z)}\!\int _{x_{1}(z,y)}^{x_{2}(z,y)}\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc340f11e458ade092393d5dacd61d2b417085b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:31.964ex; height:6.676ex;" alt="{\displaystyle V=\int _{z_{1}}^{z_{2}}\!\int _{y_{1}(z)}^{y_{2}(z)}\!\int _{x_{1}(z,y)}^{x_{2}(z,y)}\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z}" /></span></dd></dl> <p>Per la linealitat de l'integracion, un domèni dificil de definit se pòt partir en diverses susdomènis exprimables per de condicions simples. </p> <div class="mw-heading mw-heading3"><h3 id="Coordenadas_non_cartesianas">Coordenadas non cartesianas</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Volum&veaction=edit&section=6" title="Modificar la seccion : Coordenadas non cartesianas" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Volum&action=edit&section=6" title="Editar lo còdi de la seccion seguenta: Coordenadas non cartesianas"><span>modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Se lo domèni <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 {\mathcal {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}" /></span> s'exprima melhor en <a href="/w/index.php?title=Coordenadas_cilindricas&action=edit&redlink=1" class="new" title="Coordenadas cilindricas (la pagina existís pas)">coordenadas cilindricas</a> per las condicions simplas <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 {\mathcal {A}}'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdddb9dd6686547ae9ee0c91b3c27220a0e1405c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.603ex; height:2.509ex;" alt="{\displaystyle {\mathcal {A}}'}" /></span>, lo calcul es: </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 V=\iiint _{{\mathcal {A}}'}r\,\mathrm {d} r\,\mathrm {d} \theta \,dz}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <msub> <mo>∭</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> <mo>′</mo> </msup> </mrow> </msub> <mi>r</mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>r</mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=\iiint _{{\mathcal {A}}'}r\,\mathrm {d} r\,\mathrm {d} \theta \,dz}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4d1dd49c3d8db85e0716f67bb2547217decf338" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.33ex; height:5.843ex;" alt="{\displaystyle V=\iiint _{{\mathcal {A}}'}r\,\mathrm {d} r\,\mathrm {d} \theta \,dz}" /></span> onte <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 {\mathcal {A}}'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdddb9dd6686547ae9ee0c91b3c27220a0e1405c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.603ex; height:2.509ex;" alt="{\displaystyle {\mathcal {A}}'}" /></span> es una partida limitada 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 \mathbb {R} _{+}\times [0,2\pi ]\times \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>×</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">]</mo> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} _{+}\times [0,2\pi ]\times \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d13f24eac6d98e8c5e8a89d3965c20202192560b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.532ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} _{+}\times [0,2\pi ]\times \mathbb {R} }" /></span></dd></dl> <p>Se lo domèni <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 {\mathcal {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}" /></span> s'exprima melhor dins las <a href="/w/index.php?title=Coordenadas_esfericas&action=edit&redlink=1" class="new" title="Coordenadas esfericas (la pagina existís pas)">coordenadas esfericas</a> per las condicions simplas <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 {\mathcal {A}}''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/835ad27384dee235821ce3eb5a9df15f3f86b90c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.055ex; height:2.509ex;" alt="{\displaystyle {\mathcal {A}}''}" /></span>, lo calcul dona: </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 V=\iiint _{{\mathcal {A}}''}r^{2}\sin(\phi )\,\mathrm {d} r\,\mathrm {d} \theta \,\mathrm {d} \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <msub> <mo>∭</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> <mo>″</mo> </msup> </mrow> </msub> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>r</mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=\iiint _{{\mathcal {A}}''}r^{2}\sin(\phi )\,\mathrm {d} r\,\mathrm {d} \theta \,\mathrm {d} \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6957a75d4ab66711367a8f3ce0c0066e42c48505" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.563ex; height:5.843ex;" alt="{\displaystyle V=\iiint _{{\mathcal {A}}''}r^{2}\sin(\phi )\,\mathrm {d} r\,\mathrm {d} \theta \,\mathrm {d} \phi }" /></span> onte <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 {\mathcal {A}}''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/835ad27384dee235821ce3eb5a9df15f3f86b90c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.055ex; height:2.509ex;" alt="{\displaystyle {\mathcal {A}}''}" /></span> es una partida limitada 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 \mathbb {R} _{+}\times [0,2\pi ]\times [0,\pi ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>×</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">]</mo> <mo>×</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>π</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} _{+}\times [0,2\pi ]\times [0,\pi ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c0a5c85f8b7b75ead47eba32f384af500f3cdc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.676ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} _{+}\times [0,2\pi ]\times [0,\pi ]}" /></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Solid_de_rotacion">Solid de rotacion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Volum&veaction=edit&section=7" title="Modificar la seccion : Solid de rotacion" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Volum&action=edit&section=7" title="Editar lo còdi de la seccion seguenta: Solid de rotacion"><span>modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dins lo cas que lo domèni <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 {\mathcal {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}" /></span> es un solid de revolucion la frontièra qu'es engendrada per la rotacion d'una corba d'equacion <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=f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2311a6a75c54b0ea085a381ba472c31d59321514" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.672ex; height:2.843ex;" alt="{\displaystyle y=f(x)}" /></span> a l'entorn de l'axe <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 (Ox)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>O</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (Ox)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ac5db1a6b5fbda46c2b4a2a78910ca99ea75bd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.912ex; height:2.843ex;" alt="{\displaystyle (Ox)}" /></span>, lo calcul del volum se reduch a una integrala simple </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 V=\pi \int _{x_{1}}^{x_{2}}f^{2}(x)\,\mathrm {d} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mi>π</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=\pi \int _{x_{1}}^{x_{2}}f^{2}(x)\,\mathrm {d} x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/824702a0386a1e1af646092298f93e96cef4d053" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.983ex; height:6.176ex;" alt="{\displaystyle V=\pi \int _{x_{1}}^{x_{2}}f^{2}(x)\,\mathrm {d} x}" /></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Teorèma_de_la_divergéncia"><span id="Teor.C3.A8ma_de_la_diverg.C3.A9ncia"></span>Teorèma de la divergéncia</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Volum&veaction=edit&section=8" title="Modificar la seccion : Teorèma de la divergéncia" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Volum&action=edit&section=8" title="Editar lo còdi de la seccion seguenta: Teorèma de la divergéncia"><span>modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>El <a href="/w/index.php?title=Teorema_de_la_diverg%C3%A8ncia&action=edit&redlink=1" class="new" title="Teorema de la divergència (la pagina existís pas)">teorema de la divergència</a> permet reduir el càlcul del volum a una <a href="/w/index.php?title=Integrala_de_superf%C3%ADcia&action=edit&redlink=1" class="new" title="Integrala de superfícia (la pagina existís pas)">integrala de superfícia</a>: </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 V=\iiint _{A}\mathrm {d} V={\frac {1}{3}}\iint _{\partial {\mathcal {A}}}(x,y,z){\vec {n}}\,\mathrm {d} S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <msub> <mo>∭</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <msub> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=\iiint _{A}\mathrm {d} V={\frac {1}{3}}\iint _{\partial {\mathcal {A}}}(x,y,z){\vec {n}}\,\mathrm {d} S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c59387b1f2a0aef50e21079c910bb4ba8ee58186" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.053ex; height:5.843ex;" alt="{\displaystyle V=\iiint _{A}\mathrm {d} V={\frac {1}{3}}\iint _{\partial {\mathcal {A}}}(x,y,z){\vec {n}}\,\mathrm {d} S}" /></span></dd></dl> <p>Onte <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 \partial {\mathcal {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial {\mathcal {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9d0dd582162dcbed4967cf44733cd119fcd15df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.221ex; height:2.343ex;" alt="{\displaystyle \partial {\mathcal {A}}}" /></span> es la frontièra 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 {\mathcal {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}" /></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 {\vec {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49569db585c1b6306d5ffd91161775f67235fae0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:2.343ex;" alt="{\displaystyle {\vec {n}}}" /></span> lo <a href="/w/index.php?title=Vector_unitari&action=edit&redlink=1" class="new" title="Vector unitari (la pagina existís pas)">vector unitari</a> normal 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 \mathrm {d} S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72ba425d7d7a0f229457dea3c0be4a47ea303cc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.792ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} S}" /></span> dirigit cap a l'exterior 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 {\mathcal {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}" /></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Referéncias"><span id="Refer.C3.A9ncias"></span>Referéncias</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Volum&veaction=edit&section=9" title="Modificar la seccion : Referéncias" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Volum&action=edit&section=9" title="Editar lo còdi de la seccion seguenta: Referéncias"><span>modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><span style="font-size:0.95em; font-weight:bold; color:var(--color-subtle, #54595d);" title="En anglés">(<abbr title="">en</abbr>)</span> <a rel="nofollow" class="external text" href="http://www.yourdictionary.com/volume;">Your Dictionary entry for "volume"</a></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"> <i>Cox</i> <span style="font-size:0.95em; font-weight:bold; color:var(--color-subtle, #54595d);" title="En anglés">(<abbr title="">en</abbr>)</span>Coxeter, H. S. M.: <i>Regular Polytopes</i> (Methuen and Co., 1948). Table I(i).</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><span style="font-size:0.95em; font-weight:bold; color:var(--color-subtle, #54595d);" title="En anglés">(<abbr title="">en</abbr>)</span>Chris Rorres<a rel="nofollow" class="external text" href="http://www.math.nyu.edu/~crorres/Archimedes/Tomb/Cicero.html">Tomb of Archimedes: Sources</a> ed:Courant Institute of Mathematical Sciences</span> </li> </ol> <div class="mw-heading mw-heading2"><h2 id="Vejatz_tanben">Vejatz tanben</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Volum&veaction=edit&section=10" title="Modificar la seccion : Vejatz tanben" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Volum&action=edit&section=10" title="Editar lo còdi de la seccion seguenta: Vejatz tanben"><span>modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/w/index.php?title=Volum_(termodinamica)&action=edit&redlink=1" class="new" title="Volum (termodinamica) (la pagina existís pas)">Volum (termodinamica)</a></li> <li><a href="/wiki/Espaci" title="Espaci">Espaci</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Ligams_extèrnes"><span id="Ligams_ext.C3.A8rnes"></span>Ligams extèrnes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Volum&veaction=edit&section=11" title="Modificar la seccion : Ligams extèrnes" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Volum&action=edit&section=11" title="Editar lo còdi de la seccion seguenta: Ligams extèrnes"><span>modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span style="font-size:0.95em; font-weight:bold; color:var(--color-subtle, #54595d);" title="En anglés">(<abbr title="">en</abbr>)</span><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120715025010/http://www.pythia.com.ar/?id=convol">Volume calculator</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Recuperada de « <a dir="ltr" href="https://oc.wikipedia.org/w/index.php?title=Volum&oldid=2442366">https://oc.wikipedia.org/w/index.php?title=Volum&oldid=2442366</a> »</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Especial:Categorias" title="Especial:Categorias">Categorias de la pagina</a> : <ul><li><a href="/wiki/Categoria:Mod%C3%A8l_infobox" title="Categoria:Modèl infobox">Modèl infobox</a></li><li><a href="/wiki/Categoria:Geometria" title="Categoria:Geometria">Geometria</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Aquela pagina es estada modificada pel darrièr còp lo 1 julhet de 2024 a 15.19.</li> <li id="footer-info-copyright"><span style="white-space: normal"><a href="/wiki/Wikip%C3%A8dia:Drech_d%27autor" title="Wikipèdia:Drech d'autor">Drech d'autor</a> : Los tèxtes son disponibles jos <a rel="nofollow" class="external text" href="//creativecommons.org/licenses/by-sa/4.0/deed.fr">licéncia Creative Commons paternitat pertatge a l’identic</a> ; d’autras condicions se pòdon aplicar. 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