The definition of, and notation for, the triple integral of the function over a region are given in Table 7.1.1. The region is initially assumed to have plane boundaries parallel to the coordinate axes. This allows to be subdivided into rectangular parallelepipeds, indexed by in the -direction; in the -direction; and in the -direction. The point is in the relevant rectangular parallelepiped whose volume is . The function is evaluated at this point, and the value is multiplied by . 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
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Notation
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Table 7.1.1 The triple integral
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The symbol 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 is relaxed and it can then have curved boundaries.