Applications of Multiple Integrals

Multiple integrals compute physical quantities such as volume, surface area, mass, center of mass, and moments of inertia for objects with varying density distributions.

Volume Computations

Definition

The volume of a solid region $E \subseteq \mathbb{R}^3$ is $V = \iiint_E dV$ For a solid bounded above by $z = f(x,y)$ and below by $z = g(x,y)$ over a planar region $D$ : $V = \iint_D [f(x,y) - g(x,y)]\,dA$

ExampleVolume of a sphere

The volume of a ball $x^2 + y^2 + z^2 \leq R^2$ using spherical coordinates: $V = \int_0^{2\pi}\int_0^{\pi}\int_0^R \rho^2 \sin\phi \, d\rho\, d\phi\, d\theta = 2\pi \cdot 2 \cdot \frac{R^3}{3} = \frac{4}{3}\pi R^3$

Mass and Center of Mass

Definition

For a solid $E$ with density function $\rho(x,y,z)$ :

Mass: $M = \iiint_E \rho \, dV$
Center of mass: $\bar{x} = \frac{1}{M}\iiint_E x\rho\,dV$ , $\bar{y} = \frac{1}{M}\iiint_E y\rho\,dV$ , $\bar{z} = \frac{1}{M}\iiint_E z\rho\,dV$
Moments of inertia: $I_x = \iiint_E (y^2+z^2)\rho\,dV$ , $I_y = \iiint_E (x^2+z^2)\rho\,dV$ , $I_z = \iiint_E (x^2+y^2)\rho\,dV$

ExampleCenter of mass of a hemisphere

For a uniform hemisphere $E = \{x^2+y^2+z^2 \leq R^2, z \geq 0\}$ with $\rho = 1$ : By symmetry, $\bar{x} = \bar{y} = 0$ . For $\bar{z}$ : $M = \frac{2\pi R^3}{3}, \quad \iiint_E z\,dV = \int_0^{2\pi}\int_0^{\pi/2}\int_0^R (\rho\cos\phi)\rho^2\sin\phi\,d\rho\,d\phi\,d\theta = \frac{\pi R^4}{4}$ So $\bar{z} = \frac{\pi R^4/4}{2\pi R^3/3} = \frac{3R}{8}$ .

Surface Area

Definition

The surface area of the graph $z = f(x,y)$ over a region $D$ is $A = \iint_D \sqrt{1 + \left(\frac{\partial f}{\partial x}\right)^2 + \left(\frac{\partial f}{\partial y}\right)^2} \, dA$ More generally, for a parametric surface $\mathbf{r}(u,v) = (x(u,v), y(u,v), z(u,v))$ : $A = \iint_G \|\mathbf{r}_u \times \mathbf{r}_v\| \, du\, dv$

RemarkProbability applications

In probability theory, multiple integrals compute probabilities for continuous random vectors: $P((X,Y) \in D) = \iint_D f_{X,Y}(x,y)\,dA$ where $f_{X,Y}$ is the joint density function. This application connects multivariable calculus directly to statistics and data science.