Center of Mass for Uniform Distribution of Mass

The discrete-sum formulas of §5.1.3 assumed the mass was concentrated at a handful of distinct points. Real bulk bodies instead have their mass spread out continuously, and two closely related approaches handle this.

Uniformly distributed, but treated as many small pieces ($\Delta m$). If the mass is broken up into a large but finite number of small pieces, each of mass $\Delta m_i$, located at position $x_i$ (and similarly $y_i,z_i$), the summation formulas of §5.1.3 still apply directly, just written with a $\Delta m$ notation:

Genuinely continuous mass ($dm$). If instead the small mass element considered is taken to be infinitesimally small ($dm$, an "extremely small quantity" in the limiting sense of calculus), the discrete sums are replaced by definite integrals over the whole body:

Here $\int dm=M$, the total mass of the body, in every case. …