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On Dec 8, 2020, at 6:44 AM, John Denker via Phys-l <firstname.lastname@example.org> wrote:_______________________________________________
Setting aside typos, the key idea is this:
The center of mass is given by:
∫ X dm / ∫ dm 
pretty much by definition, where dm is an element of mass,
and X is position.
Note X can be one dimensional in the simple introductory
situation, or higher-dimensional if you want.
Given the symmetry of the situation, you can find the CM by
inspection, based on physicist's intuition and experience,
without doing the calculus. It's in the middle.
If you want to do the calculus, it's
∫ X dX / ∫ dX
since in this situation dm is proportional to dX.
Turn the crank and find that the CM is halfway between the
limits of integration ... in agreement with the aforementioned
intuition and experience.
This comes up All The Time.
Note that the same formula  is also the formula for weighted
average, where dm tells you how things get weighted. In the case
where dm = dX this reduces to a simple unweighted average.
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