[tex] \sqrt{2}/6 = \sqrt{ \frac{1}{18}} [/tex] which is the 
standard deviation when the variance 1/18, as my integration confirms.
I am not sure what you mean when you say when the integration is the // same as the short cut. An alternative method using integration is 
[tex] E(W-2/3) = \int_0^1 ( W- 2/3)2W = 0 [/tex]

PS if you want to know how I have done the integrals I have attached a txt file containing exactly what I typed.