I was answering a question on proving the parallel axis thereom for angular momentum and came across this: $$\int Yy'dm=Y\int y' dm=0$$ Where the position of the center of mass of an object is given by $(X,Y,Z)$, $(x',y',z')$ is a position relative to the centre of mass and m is the mass of the object.
My text book (Introduction to classical mechanics) says that this is due to the definition of the centre of mass. There are two things that I don't understand firstly why is $Y$ independent of mass whilst $y'$ is not? and secondly please can you explain what definition of the centre of mass they are using to get the reslut above? I really have no idea to the answer for either of these questions?