Charge Density and Electric Fields

[hatfarm]

This isn't the normal kind of homework question, I'm more hoping you can help me fill in the missing steps from this example given by our professor in the class notes. The below example is supposed to assist me in solving my homework problem, I understand everything except the answers circled in red below. He completely skips where he came up with them, and I can't really figure it out. The book has nothing related to this at all, and the lecture notes don't have anything on it. Once I have those numbers, solving everything is easy, but I cannot figure out what those numbers represent. Any guidance would be appreciated.

 

[Mindscrape]

He gets the 2x because the limits of integration are from -x to x, since the gaussian surface goes from -x to x where |x|<W/2. Similarly, the W comes about from integrating from -W/2 to W/2. Though it does look like he made a typo in the final solution, for x>W/2 the unit vector should be positive.

 

[hatfarm]

Thank you for the help. I appreciate it greatly.