3.7 Triple Integrals
Suppose that \(f\) is defined on a rectangular box \[B = \{(x,y,z) \mid a\leq x \leq b,\ c\leq y \leq d,\ r\leq z \leq s\} .\] Divide \([a,b]\) into \(l\) subintervals of width \(\Delta x\), \([c,d]\) into \(m\) of width \(\Delta y\), and \([r,s]\) into \(n\) of width \(\Delta z\). The planes through the division points cut \(B\) into \(lmn\) sub-boxes \[B_{ijk} = [x_{i-1},x_i]\times [y_{j-1},y_j] \times [z_{k-1},z_k],\] each of volume \(\Delta V = \Delta x\,\Delta y\,\Delta z\).
Choosing a sample point \((x^*_{ijk}, y^*_{ijk},z^*_{ijk})\) in each \(B_{ijk}\) gives the triple Riemann sum \[\sum ^l_{i=1} \sum ^m_{j=1} \sum ^n_{k=1} f(x^*_{ijk}, y_{ijk}^*, z^*_{ijk})\,\Delta V .\]
Definition 3.7.1. The triple integral of \(f\) over the box \(B\) is \[\iiint \limits _B f(x,y,z)\,dV = \lim _{l,m,n\to \infty } \sum ^l_{i=1} \sum ^m_{j=1} \sum ^n_{k=1} f(x^*_{ijk}, y_{ijk}^*, z^*_{ijk})\,\Delta V\] if the limit exists.
If \(f\) is a density, in mass per unit volume, the triple integral is the total mass of the solid; if \(f = 1\), it is the volume of \(B\).
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