Triple Integrals

Triple Integrals Setup

A Triple Integral extends integration to three dimensions: $\iiint_E f(x,y,z)\,dV$. It evaluates from the inside out using iterated integrals:

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

To Set Up Limits, follow these three steps carefully:

Applications & Pitfalls

Integrals compute physical properties. Volume is $\iiint_E 1\,dV$. Mass is $M = \iiint_E \rho\,dV$. Moment of inertia about the $z$-axis is $I_z = \iiint_E (x^2+y^2)\rho\,dV$.

Worked Example: For a sphere $x^{2}+y^{2}+z^{2} \leq R^{2}$, the volume is $V = \frac{4}{3}\pi R^{3}$.

Common Pitfall: Reversing the order of integration is never just swapping limits. You must re-describe the region from scratch using your picture, not the algebra.