Expresse a área da superfície dada como uma integral dupla iterada e, depois, encontre a área de superfície. A parte da superfície z = 2*x + y^2 acima da região triangular de vértices (0,0), (0,1) e 1,1.

Passo 1

Beleza, vamos resolver esse exercício juntos! O problema pede para a gente calcular a área de uma superfície definida por que está acima de um triângulo com vértices em , e . Vamos usar cálculo de integrais duplas para isso. Para resolver, vamos precisar saber como configurar uma integral dupla para a área de uma superfície e entender como descrever a região triangular de integração.

Passo 2

Primeiro, vamos lembrar como calcular a área de uma superfície acima de uma região no plano . A fórmula geral para a área de superfície é:
Aqui, e são as derivadas parciais de em relação a e , respectivamente.

Passo 3

Vamos calcular as derivadas parciais da nossa função :
Substituindo na fórmula da área:

Passo 4

Agora, precisamos descrever a região triangular onde vamos integrar. Os vértices do triângulo são , e . Isso significa que varia de 0 a 1, e para cada valor de , vai de 0 até . Então, nossa integral fica:

Passo 5

Vamos resolver a integral interna primeiro, em relação a :
Então, a área fica:

Passo 6

Para resolver essa integral, vamos fazer uma substituição. Seja , então:
A integral se transforma em:

Passo 7

Agora, vamos integrar :
Calculando e :
Então:

Resposta

A área da superfície é aproximadamente unidades quadradas.

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<html><head></head><body><div>Use coordenadas esféricas para encontrar o volume do sólido. O sólido dentro da esfera <span class="math-inline ">
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Calcule a integral iterada.
Calcule a integral iterada.
Calcule a integral iterada.