Mr. Integral
Math notes/Calculus III/Triple Integrals

Density, Centers of Mass and Moments of Inertia — Cheat sheet

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Core rule

m=∭δdV,xˉ=∭xδdVm,Iz=∭(x2+y2)δdV.m=\iiint\delta dV,\quad \bar x=\frac{\iiint x\delta dV}{m},\quad I_z=\iiint(x^2+y^2)\delta dV.

Watch for

Symmetry must include the density. A center uses first moments; inertia uses squared distance to an axis.