Question

Alguém desenvolve essa questão?

Answer 1

Veamos la representación de las 6 superficies.

$r_1(u,v)=(u,v,0)\\ r_2(u,v)=(u,v,1)\\ \\ r_3(u,v)=(0,u,v)\\ r_4(u,v)=(1,u,v)\\\\ r_5(u,v)=(u,0,v)\\ r_6(u,v)=(u,1,v)$

donde: $0\leq u \leq 1\;,\; 0\leq v \leq 1$

$\displaystyle \iint\limits_Sx+y+z\,dS=\sum\limits_{i=1}^6I_i$

$\displaystyle I_1=\iint\limits_{S_1} (u+v+0)\, |r_1_u\times r_1_v|\,dudv\\ \\ I_1=\int_0^1\int_0^1 u+v \;dudv\\ \\ I_1=\int_0^1\dfrac{1}{2}+v \;dv\\ \\ \boxed{I_1=1}$


$\displaystyle I_2=\iint\limits_{S_2}(u+v+1)|r_2_u\times r_2_v|\;dudv\\ \\ I_2=\int_0^1\int_0^1 u+v+1\;dudv\\ \\ \boxed{I_2=2}$


$\displaystyle I_3=\iint\limits_{S_2}(0+u+v)|r_3_u\times r_3_v|\;dudv\\ \\ I_3=\int_0^1\int_0^1u+v \, dudv\\ \\ \boxed{I_3=1}$


$\displaystyle I_4=\iint\limits_{S_2}(1+u+v)|r_4_u\times r_4_v|\;dudv\\ \\ I_4=\int_0^1\int_0^11+u+v \, dudv\\ \\ \boxed{I_4=2}$


$\displaystyle I_5=\iint\limits_{S_2}(u+0+v)|r_5_u\times r_5_v|\;dudv\\ \\ I_5=\int_0^1\int_0^1u+v \, dudv\\ \\ \boxed{I_5=1}$


$\displaystyle I_6=\iint\limits_{S_2}(u+1+v)|r_6_u\times r_6_v|\;dudv\\ \\ I_6=\int_0^1\int_0^1u+1+v \, dudv\\ \\ \boxed{I_6=2}$


$\displaystyle \boxed{\boxed{\iint\limits_Sx+y+z\,dS= 9}}$