Triple Integrals
Cheat sheet.

The key rules, formulas and reminders from all 3 topics, gathered into reference cards.

Key formulas, conditions and traps · Read down each column.

Triple Integrals and Solid Bounds

Project the solid, then integrate between its lower and upper surfaces. $\iiint_E1\,dV$ is volume.

An inner bound may depend on outer variables, but not the other way around. Reversing order requires re-describing the solid.

Cylindrical: $dV=r\,dr\,d\theta\,dz$ . Spherical with $\phi$ from $+z$ : $dV=\rho^2\sin\phi\,d\rho\,d\phi\,d\theta$ .

State the angle convention. Transform the solid, integrand and volume factor together.

m=\iiint\delta dV,\quad \bar x=\frac{\iiint x\delta dV}{m},\quad I_z=\iiint(x^2+y^2)\delta dV.

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

2 reference blocks

Project the solid, then integrate between its lower and upper surfaces. $\iiint_E1\,dV$ is volume.

An inner bound may depend on outer variables, but not the other way around. Reversing order requires re-describing the solid.

2 reference blocks

Cylindrical: $dV=r\,dr\,d\theta\,dz$ . Spherical with $\phi$ from $+z$ : $dV=\rho^2\sin\phi\,d\rho\,d\phi\,d\theta$ .

State the angle convention. Transform the solid, integrand and volume factor together.

2 reference blocks

m=\iiint\delta dV,\quad \bar x=\frac{\iiint x\delta dV}{m},\quad I_z=\iiint(x^2+y^2)\delta dV.

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