TEX document #2

The mass of the matter in a cube centered at

r⃗ = ⟨x, y, z⟩ = xî + yĵ + zk̂

is the sum of the masses of the particles in the cube. The average density in the cube is its mass over its volume:

\frac{M_{cube}}{ΔxΔyΔz} = \frac{M_{cube}}{(Δx)^{3}} .

Take the limit as the cube shrinks to the point r⃗ :

ρ(r⃗) = lim_{Δx→0} \frac{M_{_{cube at} r^{}}}{(Δx)^{3}} .

This is the density at r⃗.

Now we think of the total mass as the sum of the masses in all the cubes. In the limit of infinitesimal cubes it’s

M = ∫∫∫_{E} ρ(r⃗) dV = ∫∫∫_{E} ρ(r⃗) dx dy dz,

where E is the region in 3-space occupied by the body.

[Ultimately should insert a link, “For more details click here”, leading to a Riemann-sum discussion of this integral.]