Enunciado

Calcule a integral interada.

Passo 1

Falaaaa ai pessoal tudo blz?

Dado a integral ,

int_0^1 int_0^x int_0^(x + y) ( x + y + z ) d z d y d x

Passo 2

Resovendo a integral temos,

int_0^1 int_0^x x z + y z + (z^2)/(2) │_0^(x + y) d y d x

int_0^1 int_0^x x (x + y) + y (x + y) + ((x + y)^2)/(2) d y d x

int_1^2 int_0^x x^2 + y x + y x + y^2 + (( x^2 + 2 x y + y^2 ))/(2) d y d x

int_0^1 int_0^x x^2 + 2 y x + y^2 + (x^2)/(2) + x y + (y^2)/(2) d y d x

int_0^1 int_0^x (3 x^2)/(2) + 3 x y + (3 y^2)/(2) d y d x

int_0^1 (3 x^2)/(2) y + (3 x y^2)/(2) + (3 y^3)/(6) │_0^x d x

int_0^1 (3 x^2)/(2) x + (3 x x^2)/(2) + (3 x^3)/(6) = int_0^1 (7 x^3)/(2) d x = (7 x^4)/(8) │_0^1 = (7)/(8)

Resposta

(7)/(8)