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Why can't the center of mass equation be written as $\frac{1}{M}\int_{a}^{b} mdx$ because it is essentially the same thing and it is more likely that $m$ be written as a function of $x$?

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Your integral doesn't work because you won't get mass as a function of position. The mass of an infinitesimal volume $dx$ has to be infinitesimal, otherwise, the total mass of an infinite number of infinitesimals will be infinte. This means that you will be giving the density of the object as a function of position, so that $dm = \rho(x)dx$.

Also, the result of your expression is $1$, since $\int_a^b\rho(x) dx = M$. The center of mass of an object is a mass-weighted average of every position of the object. So, the expression must be a sum of positions, not masses. $$\bar{x} = \frac{1}{M}\int xdm = \frac{1}{M}\int x\rho(x)dx = \frac{\int x\rho(x)dx}{\int \rho(x)dx}$$

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