Section 15.6: Triple Integration

If \(f\) is continuous on the rectangular box \(B = [a, b] \times [c, d] \times [r, s]\), then \[ \iiint_B f(x, y, z) dV = \int_r^s \int_c^d \int_a^b f(x, y, z) dx dy dz \]

The iterated integral on the right side of Fubini’s Theorem means that we integrate first with respect to \(x\) (keeping \(y\) and \(z\) fixed), then we integrate with respect to \(y\) (keeping \(z\) fixed), and finally we integrate with respect to \(z\). There are five other possible orders in which we can integrate, all of which give the same value. For instance, if we integrate with respect to \(y\), then \(z\), and then \(x\), we have \[ \iiint_B f(x, y, z) dV = \int_a^b \int_r^s \int_c^d f(x, y, z) dy dz dx \]