4.1.1—Piecewise density functions

If a rod has linear density ρ(x), its mass on [a,b] is ∫ₐᵇρ(x)dx. A piecewise density uses the appropriate expression on each subinterval, and masses or centres require separate integrals.

Check continuity or a jump at the boundary, split the integral exactly there, then form x̄=(1/M)∫xρ(x)dx. Units of density must match the coordinate units.

If ρ=2 on 0≤x<1 and ρ=4 on 1≤x≤3, total mass is 2(1)+4(2)=10; the centre uses the corresponding weighted first moments.

A density value is not a mass, and using one formula across a discontinuity silently changes the physical model.