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On Dec 8, 2020, at 6:44 AM, John Denker via Phys-l <email@example.com> 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
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
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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