Vector calculus: simple surface integral problem

At least, it *should *be simple!

I am working my way through *div, grad, curl and all that* by H. M. Schey and have just finished the chapter on surface integrals and the divergence. I don't seem able to get the right answer for one of the problems, which suggests it should be possible to evaluate the surface integral without going through long-winded calculations.

I need to find

,

the surface integral over surface S.

The surface S is open and consists of 3 squares with side length b, forming half a cube at the origin. The 3 squares are adjacent and there is one in each of the xy, xz, and yz planes.

The vector function is:

.

My thinking is as follows: the three squares are identical, so I need only find the surface integral for one square and multiply it by 3.

I start with , the square in the xy plane. The unit normal vector, , is equal to and the surface integral becomes:

I know the surface integral should overall evaluate to zero, but repeating this for the 3 surfaces, I get

.

I'm fairly sure I've missed something here, but not sure what. Any suggestions?

Thanks for taking the time to read this.