Section 15.6: Triple Integration

Recall that if \(f(x) \ge 0\), then the single integral \(\int_a^b f(x) dx\) represents the area under the curve \(y = f(x)\) from \(a\) to \(b\), and if \(f(x, y) \ge 0\), then the double integral \(\iint_D f(x, y) dA\) represents the volume under the surface \(z = f(x, y)\) and above \(D\). The corresponding interpretation of a triple integral \(\iiint_E f(x, y, z) dV\), where \(f(x, y, z) \ge 0\), is not very useful because it would be the “hypervolume” of a four-dimensional object and, of course, that is very difficult to visualize. Nonetheless, the triple integral \(\iiint_E f(x, y, z) dV\) can be interpreted in different ways in different physical situations, depending on the physical interpretations of \(x, y, z,\) and \(f(x, y, z)\).

Let’s begin with the special case where \(f(x, y, z) = 1\) for all points in \(E\). Then the triple integral does represent the volume of \(E\): \[ V(E) = \iiint_E dV \tag{12} \] For example, you can see this in the case of a type 1 region by putting \(f(x, y, z) = 1\) in Formula 6: \[ \iiint_E 1 dV = \iint_D \left[ \int_{u_1(x,y)}^{u_2(x,y)} dz \right] dA = \iint_D [u_2(x, y) - u_1(x, y)] dA \] and from Section 15.2 we know this represents the volume that lies between the surfaces \(z = u_1(x, y)\) and \(z = u_2(x, y)\).