Calcule ∬_R^() ( x + y ) d A , onde R é a região no

(b) Exercício 12

Passo 1

Fala galera! No exercício 12 temos que:

∬_R^() f (x , y) d A = int_(- 1)^1 int_(- 1)^1 f (x , y) d y d x

∬_R^() f (x , y) d A = int_(- 1)^1 int_(- 1)^1 f (x , y) d x d y

Passo 2

Vamos resolver a integral dubla.

int_(- 1)^1 int_(- 1)^1 f (x , y) d y d x = int_(- 1)^1 int_(- 1)^1 (x + y) d y d x = int_(- 1)^1 [x y + (y^2)/(2)]_(- 1)^1 d x = int_(- 1)^1 (x * 1 + (1^2)/(2) - x * (- 1) - ((- 1)^2)/(2)) d x = int_(- 1)^1 (2 x) d x = [x^2]_(- 1)^1 = 0

Resolvendo a integral dupla restante.

int_(- 1)^1 int_(- 1)^1 f (x , y) d x d y = int_(- 1)^1 int_(- 1)^1 (x + y) d x d y = int_(- 1)^1 [(x^2)/(2) + x y]_(- 1)^1 d y = int_(- 1)^1 ((1^2)/(2) + 1 y - ((- 1)^2)/(2) - ( - 1 ) y) d y = int_(- 1)^1 (2 y) d y = [y^2]_(- 1)^1 = 0

Resposta

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