Triple integral

Definition

Let $B = [a_1, b_1] \times [a_2, b_2] \times [a_3, b_3]$ with $a_i < b_i$, and let $f \colon B \to \mathbb{R}$ be bounded. Partition each interval $[a_i, b_i]$, and let $B_1, \dots, B_N$ be the resulting closed sub-boxes, with volumes $\Delta V_1, \dots, \Delta V_N$. For points $p_k \in B_k$, the sum $S = \sum_{k=1}^{N} f(p_k) \, \Delta V_k$ is a Riemann sum of $f$. The function $f$ is Riemann integrable on $B$ if there is $I \in \mathbb{R}$ with this property. For each $\varepsilon > 0$, there is $\delta > 0$ with $|S - I| < \varepsilon$ for all Riemann sums whose sub-boxes have diameter less than $\delta$. Then $I$ is the triple integral of $f$ over $B$, written $\iiint_B f(x, y, z) \, dV$.