Why Does the Integral Equal Zero in Parallel Axis Theorem Proof?

[serverxeon]

Note: The following are taken from Physics for Scientists and Engineers 6E

http://img542.imageshack.us/img542/821/75796098.png
http://img152.imageshack.us/img152/5615/70724407.png
http://img804.imageshack.us/img804/6813/65815357.png

I don't really understand why is that integral equals to 0.

Can anyone explain? Thanks!

 

[darthfunnybot]

This illustration is not very clear about the order of what it is intending to describe with respect to the equations it asserts...
The drawing implies that the first and last Integrals of the line in question are taken from a coordinate system where the center of mass is at the point (X_CM,Y_CM).
To Prove P.A.T. They create two new variables x' and y' which are equal to (X - X_CM) and (Y - Y_CM) and this in practice essentially shifts the center of mass to the point (0,0) with respect to x' and y'. Since the last line is taken with respect to the Origin, the two middle integrals equate to, by definition of the center of mass, 0 + 0 because the center of mass is essentially at (0,0). I hope this helps.