Section 15.6: Triple Integration

All the applications of double integrals in Section 15.4 can be immediately extended to triple integrals. For example, if the density function of a solid object that occupies the region \(E\) is \(\rho(x, y, z)\), in units of mass per unit volume, at any given point \((x, y, z)\), then its mass is \[ m = \iiint_E \rho(x, y, z) dV \tag{13} \] and its moments about the three coordinate planes are \[ M_{yz} = \iiint_E x\rho(x, y, z) dV \quad M_{xz} = \iiint_E y\rho(x, y, z) dV \tag{14} \] \[ M_{xy} = \iiint_E z\rho(x, y, z) dV \] The center of mass is located at the point \((\bar{x}, \bar{y}, \bar{z})\), where \[ \bar{x} = \frac{M_{yz}}{m} \quad \bar{y} = \frac{M_{xz}}{m} \quad \bar{z} = \frac{M_{xy}}{m} \tag{15} \] If the density is constant, the center of mass of the solid is called the centroid of \(E\). The moments of inertia about the three coordinate axes are \[ I_x = \iiint_E (y^2 + z^2)\rho(x, y, z) dV \quad I_y = \iiint_E (x^2 + z^2)\rho(x, y, z) dV \tag{16} \] \[ I_z = \iiint_E (x^2 + y^2)\rho(x, y, z) dV \] As in Section 15.4, the total electric charge on a solid object occupying a region \(E\) and having charge density \(\sigma(x, y, z)\) is \[ Q = \iiint_E \sigma(x, y, z) dV \] If we have three continuous random variables \(X, Y,\) and \(Z\), their joint density function is a function of three variables such that the probability that \((X, Y, Z)\) lies in \(E\) is \[ P((X, Y, Z) \in E) = \iiint_E f(x, y, z) dV \] In particular, \[ P(a \le X \le b, c \le Y \le d, r \le Z \le s) = \int_a^b \int_c^d \int_r^s f(x, y, z) dz dy dx \] The joint density function satisfies \[ f(x, y, z) \ge 0 \quad \int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty f(x, y, z) dz dy dx = 1 \]