For a rectangular expanse \(E = [a,b] \times [c,d] \times [r,s]\) in \(\mathbf{R}^3\) and a function \(f\colon E \to \mathbf{R},\) the triple integral \(\iiint_E f \,\mathrm{d}V\) is equal to an iterated integral that can be evaluated “one variable at a time” by applying the fundamental theorem of calculus thrice. Per Fubini’s Theorem, if \(f\) is continuous on the expanse then the order of the differentials \(\mathrm{d}x\) and \(\mathrm{d}y\) and \(\mathrm{d}z,\) along with their bounds, may be freely permuted. \[ \iiint\limits_E f \,\mathrm{d}V \;\;=\;\; \int\limits_a^b\left(\int\limits_c^d\left(\int\limits_r^s f(x,y,z) \,\mathrm{d}z\right)\mathrm{d}y\right)\mathrm{d}x \;\;=\;\; \int\limits_a^b\int\limits_c^d\int\limits_r^s f(x,y,z) \,\mathrm{d}z\,\mathrm{d}y\,\mathrm{d}x \]
If \(E\) is not rectangular but at least has a boundary
that can be described analytically as the graphs of functions
then the triple integral \(\iiint_E f \,\mathrm{d}V\) can still be computed.
For example, if the “top and bottom” boundaries of \(E\) in the \(z\) direction per each \(xy\) region
are given by the graphs \({z = g_1(x,y)}\) and \({z = g_2(x,y)},\)
and if the “left and right” boundaries in the \(y\) direction per each \(z\) value
are given by the graphs \({y = h_1(x)}\) and \({y = h_2(x)},\)
and if the “back and front” boundaries in the \(x\) direction are \(x=a\) and \(x=b,\) then
\[
\iiint\limits_E f \,\mathrm{d}V
\;\;=\;\;
\int\limits_a^b\int\limits_{h_1(x)}^{h_2(x)}\int\limits_{g_1(x,y)}^{g_2(x,y)} f(x,y,z) \,\mathrm{d}z\,\mathrm{d}y\,\mathrm{d}x
\,.
\]
Note, as an iterated integral, the function-bounds
are always expressed in terms of variables that have yet to be integrated.
And per Fubini’s Theorem, the order of integration by be changed,
but the functions that describe the bounds would also have to change.