PLEASE READ Im needing to understand HOW this problem is wor

PLEASE READ: I\'m needing to understand HOW this problem is worked, an answer alone is worthless to me. If you don\'t show HOW you arrived at your answer, I\'ll respond with an equally worthless 1 star. Thanks!

Solution

for z=f(x,y),
NdS=+/- (-df/dx,-df/dy,1)
where +/- is chosen whether N is outward/inward

In our case
z=1-x-2y
NdS= (1,2,1)

when z=0, we get the segment x+2y=1 , so 0<=y<=1/2 and x=1-2y

F*NdS=(x,2y,z)*(1,2,1)=x+4y+z=2y (since x+2y+z=0 on S)

So we obtain

₀^1/2 ₀^1-2y 2y dx dy

=₀^1/2 2y(1-2y) dy

=y²-4/3y³|₀^1/2=

1/4-4/3*1/8=1/4-1/6=1/12