Triple Integration
Triple Integration 1. Introduction Let $f(x,y,z)$ be a continuous function defined on a closed and bounded region $V \subset \mathbb{R}^3$. The triple integral of $f$ over $V$ is defined as $$ \iiint_V f(x,y,z)\, dV $$ It represents the limit of Riemann sums: $$ \iiint_V f(x,y,z)\, dV = \lim_{\max \Delta V_i \to 0} \sum f(x_i,y_i,z_i)\,\Delta V_i $$ provided the limit exists. If $f(x,y,z)=1$, then the triple integral reduces to: $$ \iiint_V 1\, dV = \text{Volume of } V $$ In practical computation, triple integrals are evaluated as iterated integrals: $$ \iiint_V f(x,y,z)\, dV = \int_a^b \int_{g_1(x)}^{g_2(x)} \int_{h_1(x,y)}^{h_2(x,y)} f(x,y,z)\, dz\, dy\, dx $$ The order of integration may be changed whenever convenient. # 2. Explanation Think of a double integral as adding up tiny rectangles to find the area of a region. A triple integral does the same thing in three dimensions — it adds up tiny boxes (small volumes) to find: Volume...