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The definition of, and notation for, the triple integral of the function $f\left(x\,y\,z\right)$ over a region $R$ are given in Table 7.1.1. The region $R$ is initially assumed to have plane boundaries parallel to the coordinate axes. This allows $R$ to be subdivided into rectangular parallelepipeds, indexed by $i\=1\,\dots \,u$ in the $x$direction; $j\=1\,\dots \,v$ in the $y$direction; and $k\=1\,\dots \,w$ in the $z$direction. The point $\left({x}_{i}\,{y}_{j}\,{z}_{k}\right)$ is in the relevant rectangular parallelepiped whose volume is $\mathrm{\Delta}{v}_{\mathrm{ijk}}$. The function $f$ is evaluated at this point, and the value is multiplied by $\mathrm{\Delta}{v}_{\mathrm{ijk}}$. The sum of all such products is the Riemann sum, which, in the limit as the number of subdivisions becomes infinite (and the volumes shrink to zero) is the value of the triple integral.
Definition

Notation

$\underset{\left(u\,v\,w\right)\to \infty}{\mathrm{lim}}\sum _{i\=1}^{u}\sum _{jequals;1}^{v}\sum _{kequals;1}^{w}f\left({x}_{i}comma;{y}_{j}comma;{z}_{k}\right)\mathrm{Delta;}{v}_{\mathrm{ijk}}$

$\int \int {\int}_{R}f\left(x\,y\,z\right)\mathrm{dv}$

Table 7.1.1 The triple integral



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The symbol $\mathrm{dv}$ appearing in the notation for the triple integral in Table 7.1.1 represents the "element of volume" that is articulated when the triple integral is iterated.
Eventually, the restriction on $R$ is relaxed and it can then have curved boundaries.