On each side of the cube, you have to use the outward pointing normal vector. I've including those in the drawing.
Hint: There are a couple of ways to speed this problem up. Here's my advice: choose simple a parametrization for each side. If you make it simple enough, the cross product of the partials will be a unit vector in the x, y, or z direction. (In other words, i, j, or k.) That will make your life easier.
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