Vector Calculus: An Introduction

Suppose that $D\subset {\mathbb{R}}^3$ is a closed bounded set, and $f:D\to \mathbb{R}$ a continuous function. We now subdivide $D$ into small rectangular boxes, either approaching $D$ from the in- or the outside, similarly as shown in Figure 5.7 for plane domains. Suppose the partition has $n$ boxes. Number them in some order from $1$ to $n\text{,}$ and denote the $i$-th box by $R_i\text{.}$ These boxes then have all the form

and their volume is $\mathrm{\Delta }{x}_i\mathrm{\Delta }{y}_i\mathrm{\Delta }{z}_i\text{.}$ We then choose an arbitrary point $(x_i^{\ast },y_i^{\ast },z_i^{\ast })\in R_i$ and consider the sum

which is the counterpart of the Riemann sum (5.1). If $D$ is `reasonably’ smooth and $f$ is continuous then one can show that the above sum, also called a Riemann sum, has a limit as $\underset{i}{max}\{\mathrm{\Delta }{x}_i,\mathrm{\Delta }{y}_i,\mathrm{\Delta }{z}_i\}\to 0\text{.}$ Moreover, the limit is independent of the choice of $(x_i^{\ast },y_i^{\ast },z_i^{\ast })\in R_i\text{.}$ The limit is denoted by

or in more modern notation simply by

and called the (triple) integral of $f$ over $D$.