What is the Explanation for the Zero Value in the Steiner Theorem Demonstration?

[Telemachus]

Homework Statement

I have a doubt about the steiner theorem demonstration, its actually trivial, but I can't realize why is this.

Lets see, the demonstration which you can see http://en.wikipedia.org/wiki/Parallel_axis_theorem" goes as follows:

$$I_{cm} = \int{(x^2 + y^2)} dm$$

$$I_z = \int{((x - r)^2 + y^2)} dm$$

$$I_z = \int{(x^2 + y^2)} dm + r^2 \int dm - 2r\int{x} dm$$

$$I_z = I_{cm} + mr^2$$

What I need to know is why this gives zero:
$$2r\int{x} dm$$

Bye there, and thanks for your help :)

 

[Doc Al]

What I need to know is why this gives zero:
$$2r\int{x} dm$$

Because they are using a coordinate system in which the center of mass is at the origin, so:
Xcm ≡ (∫xdm)/M = 0

 

[Telemachus]

I don't get it. I actually read that explanation before. That integral wouldn't give: $$2rxM$$? being M the total mass...? I'm not seeing the "differential" thing and how it works to give zero, I think that's the problem. It must be like integrating over a null area, but I don't see it.

 

[Doc Al]

I don't get it. I actually read that explanation before. That integral wouldn't give: $$2rxM$$? being M the total mass...? I'm not seeing the "differential" thing and how it works to give zero, I think that's the problem. It must be like integrating over a null area, but I don't see it.

Do you agree that the x-coordinate of the center of mass is given by:

$$x_{cm} = \frac{1}{M} \int x dm$$

where M is the total mass?

 

[Telemachus]

Right. I see it now :P thanks.