Calcule a integral dupla. $\iint_{D} x y d A, D$ é a região triangular com vértices $(0,0) ;(1,2) ;(0,3)$.

Passo 1

Beleza, vamos resolver essa integral dupla juntos! O enunciado pede para calcular a integral , onde é uma região triangular com vértices em , e . Vamos usar conhecimentos de cálculo integral, especificamente sobre integrais duplas e como determinar os limites de integração para uma região triangular.

Passo 2

Primeiro, vamos entender a região . Como os vértices são , e , podemos desenhar o triângulo e identificar as linhas que formam os lados. Precisamos encontrar as equações dessas retas para determinar os limites de integração.

Passo 3

Vamos encontrar as equações das retas que ligam os vértices:
  1. De a :
    • Slope (coeficiente angular)
    • Equação:
  2. De a :
    • Slope
    • Usando o ponto :
  3. De a :
    • É uma linha vertical:

Passo 4

Agora, vamos determinar os limites de integração. Observando o triângulo:
  • varia de a .
  • Para cada , varia de até .
Então, a integral dupla fica:

Passo 5

Vamos resolver a integral interna em relação a primeiro:
Integrando :

Passo 6

Simplificando a expressão:

Passo 7

Agora, vamos integrar em relação a :
Integrando termo a termo:

Passo 8

Substituindo os limites:

Resposta

O valor da integral dupla é .

Exercícios relacionados

Calcule a integral iterada
<html><head></head><body><div>Determine a massa e o centro de massa da lâmina que ocupa a região D e tem função densidade <span class="math-inline " data-overflow=“visible”>
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Determine a massa e o centro de massa se a densidade de massa em qualquer ponto for proporcional à distância a origem.</div>
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<html><head></head><body><div>Determine o volume do sólido limitado pela superfície <span class="math-inline ">
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Determine se a série geométrica é convergente ou divergente. Se for convergente, calcule a sua soma: