I had a lab report to do based around moments of inertia and thought I'd remind myself of the parallel axis theorem, so I looked up the derivation. It goes something like:

$$ I= \int [(x+D)^2 + y^2]dm$$

$$I = \int (x^2+y^2) dm + D^2\int dm + 2D\int xdm$$

Where the last term is equal to zero. I was confused by this so I decided to look around a bit and none of the explanatioms have made sense to me. They explain that this term is equal to zero because the x component of the center of mass is zero, if we have our origin as the center of mass. I can understand that, however, wouldn't that indicate that the x squared should also be equal to zero? I suppose the problem here is me not understanding exactly what these terms are saying. Thank you in advance.