Question:Evaluate$\iint A.dS$ where $A=y\hat i+2x\hat j-z\hat k$ and S is the surface of the plane $2x+y=6$ in the first octant cut off by the plane $z=4$
My Approach:I roughly sketch and consider $5$ surfaces.
$S_1$ is the triangle in the plane $z = 0$
$S_2$ is the triangle in the plane $z = 4$
$S_3$ is the rectangle in the plane $x = 0$
$S_4$ is the rectangle in the plane $y = 0$
$S_5$ is the plane $2x+y=6$.
The normal vectors to these respective surfaces are $(0,0,-1), (0,0,1), (-1,0,0),(0,-1,0), (2,1,0) $ respectively.Then i evaluate the surfaces $S_1, S_2 ,S_3,S_4,S_5$ And Add them together.But my answer is incorrect.the solution provided by the book is correct.
I think my approaching is incorrect.Please explain me how to do this in right way.
Thanks in advance.