In Double Integrals over Rectangular Regions, we discussed the double integral of a function $f(x,y)$ of two variables over a rectangular region in the plane. In this section we define the triple integral of a function $f(x,y,z)$ of three variables over a rectangular solid box in space, $\mathbb{R}^{3}$. Later in this section we extend the definition to more general regions in $\mathbb{R}^{3}$.