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	<title>Valor médio de uma função num paralelepípedo - Histórico de revisões</title>
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	<updated>2026-09-12T20:34:01Z</updated>
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	<entry>
		<id>http://www.mysolutions.tecnico.ulisboa.pt//wiki/index.php?title=Valor_m%C3%A9dio_de_uma_fun%C3%A7%C3%A3o_num_paralelep%C3%ADpedo&amp;diff=4360&amp;oldid=prev</id>
		<title>Ist178052 em 13h43min de 4 de abril de 2018</title>
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		<updated>2018-04-04T13:43:38Z</updated>

		<summary type="html">&lt;p&gt;&lt;/p&gt;
&lt;table class=&quot;diff diff-contentalign-left diff-editfont-monospace&quot; data-mw=&quot;interface&quot;&gt;
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				&lt;td colspan=&quot;2&quot; style=&quot;background-color: #fff; color: #202122; text-align: center;&quot;&gt;← Revisão anterior&lt;/td&gt;
				&lt;td colspan=&quot;2&quot; style=&quot;background-color: #fff; color: #202122; text-align: center;&quot;&gt;Revisão das 13h43min de 4 de abril de 2018&lt;/td&gt;
				&lt;/tr&gt;&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot; id=&quot;mw-diff-left-l30&quot; &gt;Linha 30:&lt;/td&gt;
&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Linha 30:&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class='diff-marker'&gt; &lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;/td&gt;&lt;td class='diff-marker'&gt; &lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class='diff-marker'&gt; &lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;/td&gt;&lt;td class='diff-marker'&gt; &lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class='diff-marker'&gt;−&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;Para obter o zip que contém as instâncias deste exercício clique aqui&lt;del class=&quot;diffchange diffchange-inline&quot;&gt;(valorMedioR3)&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td class='diff-marker'&gt;+&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;Para obter o zip que contém as instâncias deste exercício clique aqui&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;[https://drive.tecnico.ulisboa.pt/api/drive/file/851498741306514/download]&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class='diff-marker'&gt; &lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;/td&gt;&lt;td class='diff-marker'&gt; &lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class='diff-marker'&gt; &lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;Se deseja obter o código fonte que gera os exercícios contacte miguel.dziergwa@ist.utl.pt&lt;/div&gt;&lt;/td&gt;&lt;td class='diff-marker'&gt; &lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;Se deseja obter o código fonte que gera os exercícios contacte miguel.dziergwa@ist.utl.pt&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
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		<author><name>Ist178052</name></author>
	</entry>
	<entry>
		<id>http://www.mysolutions.tecnico.ulisboa.pt//wiki/index.php?title=Valor_m%C3%A9dio_de_uma_fun%C3%A7%C3%A3o_num_paralelep%C3%ADpedo&amp;diff=4252&amp;oldid=prev</id>
		<title>Ist178052: Criou a página com &quot;&lt;div class=&quot;toccolours mw-collapsible mw-collapsed&quot; style=&quot;width:420px&quot;&gt; '''Metadata''' &lt;div class=&quot;mw-collapsible-content&quot;&gt; *CONTEXTO : Primeiro ciclo universitário *AREA:...&quot;</title>
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		<updated>2018-04-02T12:53:54Z</updated>

		<summary type="html">&lt;p&gt;Criou a página com &amp;quot;&amp;lt;div class=&amp;quot;toccolours mw-collapsible mw-collapsed&amp;quot; style=&amp;quot;width:420px&amp;quot;&amp;gt; &amp;#039;&amp;#039;&amp;#039;Metadata&amp;#039;&amp;#039;&amp;#039; &amp;lt;div class=&amp;quot;mw-collapsible-content&amp;quot;&amp;gt; *CONTEXTO : Primeiro ciclo universitário *AREA:...&amp;quot;&lt;/p&gt;
&lt;p&gt;&lt;b&gt;Página nova&lt;/b&gt;&lt;/p&gt;&lt;div&gt;&amp;lt;div class=&amp;quot;toccolours mw-collapsible mw-collapsed&amp;quot; style=&amp;quot;width:420px&amp;quot;&amp;gt;&lt;br /&gt;
'''Metadata'''&lt;br /&gt;
&amp;lt;div class=&amp;quot;mw-collapsible-content&amp;quot;&amp;gt;&lt;br /&gt;
*CONTEXTO : Primeiro ciclo universitário&lt;br /&gt;
*AREA: Matemática&lt;br /&gt;
*DISCIPLINA: Calculo Diferencial e Integral 2&lt;br /&gt;
*ANO: 1&lt;br /&gt;
*LINGUA: pt&lt;br /&gt;
*AUTOR: Ana Moura Santos e Miguel Dziergwa&lt;br /&gt;
*MATERIA PRINCIPAL: Aplicações ao cálculo de grandezas físicas&lt;br /&gt;
*DESCRICAO: Valor médio de uma função num paralelepípedo&lt;br /&gt;
*DIFICULDADE: *&lt;br /&gt;
*TEMPO MEDIO DE RESOLUCAO: 15 mn&lt;br /&gt;
*TEMPO MAXIMO DE RESOLUCAO: 30 mn&lt;br /&gt;
*PALAVRAS CHAVE: função integrável à Riemann, integral triplo, ordem de integração, extremos de integração, integral como valor médio&lt;br /&gt;
&amp;lt;/div&amp;gt;&lt;br /&gt;
&amp;lt;/div&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
O valor médio de \(f\)\(\left(\begin{array}{c}x\\y\\z\\\end{array}\right)\)=\(-3\cos(y)-5\sin(z)\) na região \(A=\)\(\left[\frac{5\pi}{6},\frac{3\pi}{2}\right]\times\left[-\frac{3\pi}{4},\frac{5\pi}{6}\right]\times\left[-\frac{2\pi}{3},\frac{\pi}{3}\right]\) de \( \mathbb{R^3} \) é igual a:&lt;br /&gt;
&lt;br /&gt;
A) \(\frac{77-18\sqrt{2}}{19\pi}\)&lt;br /&gt;
&lt;br /&gt;
B) \(\frac{1}{18}\left(77-18\sqrt{2}\right)\pi^2\)&lt;br /&gt;
&lt;br /&gt;
C) \(\frac{19\pi^3}{18}\)&lt;br /&gt;
&lt;br /&gt;
D) \(\frac{893\pi^2}{72}\)&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Para obter o zip que contém as instâncias deste exercício clique aqui(valorMedioR3)&lt;br /&gt;
&lt;br /&gt;
Se deseja obter o código fonte que gera os exercícios contacte miguel.dziergwa@ist.utl.pt&lt;/div&gt;</summary>
		<author><name>Ist178052</name></author>
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