Triple Integrals

$$\iiint_E f(x,y,z)\,dV$$

Evaluation — Iterate from outside in:

$$\int_a^b\int_{g_1}^{g_2}\int_{h_1}^{h_2} f\,dz\,dy\,dx$$

Applications

• Volume: $\iiint_E 1\,dV$
• Mass: $M = \iiint_E \rho\,dV$
• Moment of inertia ($z$-axis):

$$I_z = \iiint_E (x^2+y^2)\rho\,dV$$

Setting up limits:

1. Sketch the region $E$.
2. Project onto a coordinate plane for the outer limits.
3. For each $(x,y)$, find the $z$-range.

Example — Sphere $x^{2}+y^{2}+z^{2} \leq R^{2}$: $V = \frac{4}{3}\pi R^{3}$.

Tip: If the region has symmetry, switch coordinates before integrating.

Common pitfall: Reversing the order of integration is not swapping the limit expressions — the region must be re-described from scratch. The picture, not the algebra, dictates the new limits.