Compute Flux of Vector Field Through Surface S

[theown1]

Homework Statement

Compute the flux of the vector field, , through the surface, S.
$$\vec{F}$$= 3xi + yj + zk and S is the part of the surface z + 4x + 2y = 12 in the first octant oriented upward.

Homework Equations

by definition from my book the integral is
$$\int$$F(x,y,f(x,y)$$\circ$$<-f_x,f_y,1>dxdy

for a plane oriented up

The Attempt at a Solution

So to get f(x,y) from the surface i did
z=f(x,y)=12-4x-2y

I have to find the integral of the vector field dot product with <-f_x,-f_y,1> which turns out to be <4,2,1>

So <3x,y,12-4x-2y>dot<4,2,1>=8x+12

Next I have to find
$$\int$$8x+12dxdy

I'm not sure what to set my values at to solve this, I tried 0<x<3 and 0<y<6
With those values my integral equaled 288 which was the wrong answer
then I tried to dot the first part with <-4,-2,1> and my answer was -432 which was still wrong

Can someone help me find the integration values for the first octant of that plane, I think that's all I need to solve this?

 

[Dick]

The xy domain is a triangle, it's bounded by x=0, y=0 and the line 0=12-4x-2y, isn't it? 0<x<3 and 0<y<6 is a rectangle.