For a continuous function where is a cubical region

the triple integral of over , denoted by

can be defined in a manner similar to that of the double integral. We divide the region into rectangular cells where the volume of the th cell is We select any arbitrary point and consider the limit of the (Riemann) sums

the limit being taken as with The triple integral is this limit, whenever it exists. As in the two variables case, this can be extended to closed bounded regions We omit the details. The triple integral is also written as

Triple integral satisfies properties similar to the ones satisfied by double integrals, see theorem 40.1.3.

The evaluation of triple integrals becomes possible because of the three dimensional version of Fubini's theorem.