The Applications of Multiple Integrals

The real value of a multiple integral is that it unifies the solutions to three kinds of problem — geometric, physical and probabilistic: regard the integrand as a density or a weight, and the integral gives some total quantity.

In geometry it is used to find areas, volumes and the mass distribution over a surface; in physics it is used to find centres of mass, moments of inertia and gravitational forces — these three look alike because they are all “the integral of a density times some quantity”, and only the weight differs. In probability, the definite integral is the distribution function of a continuous random variable, and the multiple integral is the generalisation of the joint distribution.

Once you have this unified viewpoint, the multiple integral is no longer an isolated computational technique but a general language for handling problems of “accumulating continuous quantities”.