Integral em coordenadas cartesianas

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Metadata

  • CONTEXTO : Primeiro ciclo universitário
  • AREA: Matemática
  • DISCIPLINA: Calculo diferencial e integral 2
  • ANO: 1
  • LINGUA: pt
  • AUTOR: Ana Moura Santos e Miguel Dziergwa
  • MATERIA PRINCIPAL: Integrais múltiplos: Teorema de Fubini
  • DESCRICAO: Integral em coordenadas cartesianas
  • DIFICULDADE: ***
  • TEMPO MEDIO DE RESOLUCAO: 15 mn
  • TEMPO MAXIMO DE RESOLUCAO: 20 mn
  • PALAVRAS CHAVE: integral triplo, ordem de integração, extremos de integração, coordenadas cartesianas

Sendo f uma função integrável e positiva, a soma de integrais triplos iterados \(\begin{array}{c}\int_0^1\int_z^1\int_0^1\text{f}\left(\begin{array}{c}x\\y\\z\\\end{array}\right)dxdydz\\+\int_1^2\int_0^1\int_{2-x}^1\text{f}\left(\begin{array}{c}x\\y\\z\\\end{array}\right)dzdydx\\+\int_0^1\int_0^{1-y}\int_0^{-y-z+1}\text{f}\left(\begin{array}{c}x\\y\\z\\\end{array}\right)dxdzdy\\\end{array}\) pode também ser dada, após uma mudança da ordem de integração, por

A)\(\fbox{$\begin{array}{c}\int_1^2\int_1^2\int_{3-x}^2\text{f}\left(\begin{array}{c}x\\y\\z\\\end{array}\right)dzdydx\\+\int_1^2\int_1^{3-x}\int_1^{-x-y+4}\text{f}\left(\begin{array}{c}x\\y\\z\\\end{array}\right)dzdydx\\+\int_1^2\int_{3-x}^2\int_1^2\text{f}\left(\begin{array}{c}x\\y\\z\\\end{array}\right)dydzdx\\\end{array}$}\)

B)\(\fbox{$\begin{array}{c}\int_1^2\int_{3-x}^2\int_1^2\text{f}\left(\begin{array}{c}x\\y\\z\\\end{array}\right)dydzdx\\+\int_1^2\int_1^{3-z}\int_1^{-y-z+4}\text{f}\left(\begin{array}{c}x\\y\\z\\\end{array}\right)dxdydz\\+\int_1^2\int_{3-z}^2\int_1^2\text{f}\left(\begin{array}{c}x\\y\\z\\\end{array}\right)dydxdz\\\end{array}$}\)

C)\(\fbox{$\begin{array}{c}\int_1^2\int_{3-x}^2\int_1^2\text{f}\left(\begin{array}{c}x\\y\\z\\\end{array}\right)dydzdx\\+\int_1^2\int_1^{3-x}\int_1^{-x-y+4}\text{f}\left(\begin{array}{c}x\\y\\z\\\end{array}\right)dzdydx\\+\int_1^2\int_1^y\int_1^2\text{f}\left(\begin{array}{c}x\\y\\z\\\end{array}\right)dxdzdy\\\end{array}$}\)

D)\(\fbox{$\begin{array}{c}\int_1^2\int_1^2\int_z^1\text{f}\left(\begin{array}{c}x\\y\\z\\\end{array}\right)dydzdx\\+\int_1^2\int_1^2\int_{3-x}^2\text{f}\left(\begin{array}{c}x\\y\\z\\\end{array}\right)dzdxdy\\+\int_1^2\int_1^2\int_1^y\text{f}\left(\begin{array}{c}x\\y\\z\\\end{array}\right)dzdxdy\\\end{array}$}\)

E)Nenhuma das anteriores

Para obter o zip que contém as instâncias deste exercício clique aqui[1]

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