Coordenadas polares - Histórico de revisões

Ist12543 em 20h51min de 23 de março de 2018

2018-03-23T20:51:20Z

Ist12543 em 18h46min de 19 de fevereiro de 2018

2018-02-19T18:46:07Z

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2016-08-30T10:13:02Z

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2016-08-30T09:58:27Z

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*CONTEXTO : Primeiro ciclo universitário
*AREA: Matemática
*DISCIPLINA: Calculo diferencial e integral 2
*ANO: 1
*LINGUA: pt
*AUTOR: Equipa Calculo diferencial e integral 2
*MATERIA PRINCIPAL:
*DESCRICAO:
*DIFICULDADE: easy
*TEMPO MEDIO DE RESOLUCAO: 15 mn
*TEMPO MAXIMO DE RESOLUCAO: 30 mn
*PALAVRAS CHAVE:
</div>
</div>

Sendo f uma função positiva e integrável, a seguinte soma de integrais em coordenadas polares $\begin{array}{c}\text{}\int_0^1\int_0^{\frac{5\pi}{4}}r\text{f}\left(\begin{array}{c}r\cos(\theta)\\r\sin(\theta)\\\end{array}\right)d\thetadr\\+\int_1^{\sqrt{2}}\int_{\frac{\pi}{4}}^{\frac{5\pi}{4}}r\text{f}\left(\begin{array}{c}r\cos(\theta)\\r\sin(\theta)\\\end{array}\right)d\thetadr\\+\int_{\sqrt{2}}^2\int_{-\arccos\left(-\frac{\sqrt{2}}{r}\right)}^{-\frac{3\pi}{4}}r\text{f}\left(\begin{array}{c}r\cos(\theta)\\r\sin(\theta)\\\end{array}\right)d\thetadr\\\end{array}$ pode também ser dada por:

A)$\fbox{$\begin{array}{c}\text{}\int_0^1\int_0^{\frac{5\pi}{4}}r\text{f}\left(\begin{array}{c}r\cos(\theta)\\r\sin(\theta)\\\end{array}\right)d\thetadr\\+\int_1^{\sqrt{2}}\int_{\arccos\left(\frac{1}{r}\right)}^{\frac{5\pi}{4}}r\text{f}\left(\begin{array}{c}r\cos(\theta)\\r\sin(\theta)\\\end{array}\right)d\thetadr\\+\int_{\sqrt{2}}^2\int_{-\arccos\left(-\frac{\sqrt{2}}{r}\right)}^{-\frac{3\pi}{4}}r\text{f}\left(\begin{array}{c}r\cos(\theta)\\r\sin(\theta)\\\end{array}\right)d\thetadr\\\end{array}$}$

B)$\fbox{$\begin{array}{c}\text{}\int_0^1\int_0^{\frac{5\pi}{4}}r\text{f}\left(\begin{array}{c}r\cos(\theta)\\r\sin(\theta)\\\end{array}\right)d\thetadr\\+\int_1^{\sqrt{2}}\int_{\arccos\left(\frac{1}{r}\right)}^{\frac{5\pi}{4}}r\text{f}\left(\begin{array}{c}r\cos(\theta)\\r\sin(\theta)\\\end{array}\right)d\thetadr\\+\int_{\sqrt{2}}^2\int_{-\arccos\left(-\frac{\sqrt{2}}{r}\right)}^{-\frac{3\pi}{4}}r\text{f}\left(\begin{array}{c}r\cos(\theta)\\r\sin(\theta)\\\end{array}\right)d\thetadr\\\end{array}$}$

C)$\fbox{$\begin{array}{c}\text{}\int_0^1\int_0^{\frac{5\pi}{4}}r\text{f}\left(\begin{array}{c}r\cos(\theta)\\r\sin(\theta)\\\end{array}\right)d\thetadr\\+\int_1^{\sqrt{2}}\int_{\arccos\left(\frac{1}{r}\right)}^{\frac{5\pi}{4}}r\text{f}\left(\begin{array}{c}r\cos(\theta)\\r\sin(\theta)\\\end{array}\right)d\thetadr\\+\int_{\sqrt{2}}^2\int_{-\arccos\left(-\frac{\sqrt{2}}{r}\right)}^{-\frac{3\pi}{4}}r\text{f}\left(\begin{array}{c}r\cos(\theta)\\r\sin(\theta)\\\end{array}\right)d\thetadr\\\end{array}$}$

D)Nenhuma das anteriores

Para obter o zip que contém as instâncias deste exercício clique aqui(coordPolaCartes)

Se deseja obter o código fonte que gera os exercícios contacte miguel.dziergwa@ist.utl.pt

@@ Linha 4: / Linha 4: @@
 *CONTEXTO : Primeiro ciclo universitário
 *AREA: Matemática
-*DISCIPLINA: Calculo diferencial e integral 2
+*DISCIPLINA: Calculo Diferencial e Integral 2
 *ANO: 1
 *LINGUA: pt
-*AUTOR: Equipa Calculo diferencial e integral 2
+*AUTOR: Ana Moura Santos e Miguel Dziergwa
-*MATERIA PRINCIPAL: Integral duplo
+*MATERIA PRINCIPAL: Teorema de mudança de variáveis
 *DESCRICAO:  Integral duplo em coordenadas polares
-*DIFICULDADE: easy
+*DIFICULDADE: ***
 *TEMPO MEDIO DE RESOLUCAO: 15 mn
 *TEMPO MAXIMO DE RESOLUCAO: 30 mn
-*PALAVRAS CHAVE:  integral múltiplo, ordem de integração, extremos de integração, coordenadas polares
+*PALAVRAS CHAVE:  integral duplo, ordem de integração, extremos de integração, coordenadas polares
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 </div>

@@ Linha 17: / Linha 17: @@
 </div>
-Sendo f uma função positiva e integrável, a seguinte soma de integrais em coordenadas polares \(\begin{array}{c}\text{}\int_0^1\int_0^{\frac{5\pi}{4}}r\text{f}\left(\begin{array}{c}r\cos(\theta)\\r\sin(\theta)\\\end{array}\right)d\thetadr\\+\int_1^{\sqrt{2}}\int_{\frac{\pi}{4}}^{\frac{5\pi}{4}}r\text{f}\left(\begin{array}{c}r\cos(\theta)\\r\sin(\theta)\\\end{array}\right)d\thetadr\\+\int_{\sqrt{2}}^2\int_{-\arccos\left(-\frac{\sqrt{2}}{r}\right)}^{-\frac{3\pi}{4}}r\text{f}\left(\begin{array}{c}r\cos(\theta)\\r\sin(\theta)\\\end{array}\right)d\thetadr\\\end{array}\) pode também ser dada por:
+Sendo f uma função positiva e integrável, a seguinte soma de integrais em coordenadas polares \(\begin{array}{c}\text{}\int_0^1\int_0^{\frac{5\pi}{4}}r\text{f}\left(\begin{array}{c}r\cos(\theta)\\r\sin(\theta)\\\end{array}\right)d\theta dr\\+\int_1^{\sqrt{2}}\int_{\frac{\pi}{4}}^{\frac{5\pi}{4}}r\text{f}\left(\begin{array}{c}r\cos(\theta)\\r\sin(\theta)\\\end{array}\right)d\theta dr\\+\int_{\sqrt{2}}^2\int_{-\arccos\left(-\frac{\sqrt{2}}{r}\right)}^{-\frac{3\pi}{4}}r\text{f}\left(\begin{array}{c}r\cos(\theta)\\r\sin(\theta)\\\end{array}\right)d\theta dr\\\end{array}\) pode também ser dada por:
-A)\(\fbox{$\begin{array}{c}\text{}\int_0^1\int_0^{\frac{5\pi}{4}}r\text{f}\left(\begin{array}{c}r\cos(\theta)\\r\sin(\theta)\\\end{array}\right)d\thetadr\\+\int_1^{\sqrt{2}}\int_{\arccos\left(\frac{1}{r}\right)}^{\frac{5\pi}{4}}r\text{f}\left(\begin{array}{c}r\cos(\theta)\\r\sin(\theta)\\\end{array}\right)d\thetadr\\+\int_{\sqrt{2}}^2\int_{-\arccos\left(-\frac{\sqrt{2}}{r}\right)}^{-\frac{3\pi}{4}}r\text{f}\left(\begin{array}{c}r\cos(\theta)\\r\sin(\theta)\\\end{array}\right)d\thetadr\\\end{array}$}\)
+A)\(\fbox{$\begin{array}{c}\text{}\int_0^1\int_0^{\frac{5\pi}{4}}r\text{f}\left(\begin{array}{c}r\cos(\theta)\\r\sin(\theta)\\\end{array}\right)d\theta dr\\+\int_1^{\sqrt{2}}\int_{\arccos\left(\frac{1}{r}\right)}^{\frac{5\pi}{4}}r\text{f}\left(\begin{array}{c}r\cos(\theta)\\r\sin(\theta)\\\end{array}\right)d\theta dr\\+\int_{\sqrt{2}}^2\int_{-\arccos\left(-\frac{\sqrt{2}}{r}\right)}^{-\frac{3\pi}{4}}r\text{f}\left(\begin{array}{c}r\cos(\theta)\\r\sin(\theta)\\\end{array}\right)d\theta dr\\\end{array}$}\)
-B)\(\fbox{$\begin{array}{c}\text{}\int_0^1\int_0^{\frac{5\pi}{4}}r\text{f}\left(\begin{array}{c}r\cos(\theta)\\r\sin(\theta)\\\end{array}\right)d\thetadr\\+\int_1^{\sqrt{2}}\int_{\arccos\left(\frac{1}{r}\right)}^{\frac{5\pi}{4}}r\text{f}\left(\begin{array}{c}r\cos(\theta)\\r\sin(\theta)\\\end{array}\right)d\thetadr\\+\int_{\sqrt{2}}^2\int_{-\arccos\left(-\frac{\sqrt{2}}{r}\right)}^{-\frac{3\pi}{4}}r\text{f}\left(\begin{array}{c}r\cos(\theta)\\r\sin(\theta)\\\end{array}\right)d\thetadr\\\end{array}$}\)
+B)\(\fbox{$\begin{array}{c}\text{}\int_0^1\int_0^{\frac{5\pi}{4}}r\text{f}\left(\begin{array}{c}r\cos(\theta)\\r\sin(\theta)\\\end{array}\right)d\theta dr\\+\int_1^{\sqrt{2}}\int_{\arccos\left(\frac{1}{r}\right)}^{\frac{5\pi}{4}}r\text{f}\left(\begin{array}{c}r\cos(\theta)\\r\sin(\theta)\\\end{array}\right)d\theta dr\\+\int_{\sqrt{2}}^2\int_{-\arccos\left(-\frac{\sqrt{2}}{r}\right)}^{-\frac{3\pi}{4}}r\text{f}\left(\begin{array}{c}r\cos(\theta)\\r\sin(\theta)\\\end{array}\right)d\theta dr\\\end{array}$}\)
-C)\(\fbox{$\begin{array}{c}\text{}\int_0^1\int_0^{\frac{5\pi}{4}}r\text{f}\left(\begin{array}{c}r\cos(\theta)\\r\sin(\theta)\\\end{array}\right)d\thetadr\\+\int_1^{\sqrt{2}}\int_{\arccos\left(\frac{1}{r}\right)}^{\frac{5\pi}{4}}r\text{f}\left(\begin{array}{c}r\cos(\theta)\\r\sin(\theta)\\\end{array}\right)d\thetadr\\+\int_{\sqrt{2}}^2\int_{-\arccos\left(-\frac{\sqrt{2}}{r}\right)}^{-\frac{3\pi}{4}}r\text{f}\left(\begin{array}{c}r\cos(\theta)\\r\sin(\theta)\\\end{array}\right)d\thetadr\\\end{array}$}\)
+C)\(\fbox{$\begin{array}{c}\text{}\int_0^1\int_0^{\frac{5\pi}{4}}r\text{f}\left(\begin{array}{c}r\cos(\theta)\\r\sin(\theta)\\\end{array}\right)d\theta dr\\+\int_1^{\sqrt{2}}\int_{\arccos\left(\frac{1}{r}\right)}^{\frac{5\pi}{4}}r\text{f}\left(\begin{array}{c}r\cos(\theta)\\r\sin(\theta)\\\end{array}\right)d\theta dr\\+\int_{\sqrt{2}}^2\int_{-\arccos\left(-\frac{\sqrt{2}}{r}\right)}^{-\frac{3\pi}{4}}r\text{f}\left(\begin{array}{c}r\cos(\theta)\\r\sin(\theta)\\\end{array}\right)d\theta dr\\\end{array}$}\)
 D)Nenhuma das anteriores

Coordenadas polares - Histórico de revisões

Ist12543 em 20h51min de 23 de março de 2018

Ist12543 em 18h46min de 19 de fevereiro de 2018

Ist178052 em 10h13min de 30 de agosto de 2016

Ist178052: Criou a página com " '''Metadata''' *CONTEXTO : Primeiro ciclo universitário *AREA:..."

Ist178052: Criou a página com "
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