Some Practical Applications
Quite generally, just as in one variable, one can use the multiple integral to find the average of a function over a given set. Given a set D ⊆ Rn and an integrable function f over D, the average value of f over its domain is given by
where m(D) is the measure of D.
Additionally, multiple integrals are used in many applications in physics. The examples below also show some variations in the notation.
In mechanics, the moment of inertia is calculated as the volume integral (triple integral) of the density weighed with the square of the distance from the axis:
The gravitational potential associated with a mass distribution given by a mass measure dm on three-dimensional Euclidean space R3 is
If there is a continuous function ρ(x) representing the density of the distribution at x, so that dm(x) = ρ(x)d 3x, where d 3x is the Euclidean volume element, then the gravitational potential is
In electromagnetism, Maxwell's equations can be written using multiple integrals to calculate the total magnetic and electric fields. In the following example, the electric field produced by a distribution of charges given by the volume charge density is obtained by a triple integral of a vector function:
This can also be written as an integral with respect to a signed measure representing the charge distribution.
Read more about this topic: Multiple Integral
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“Great statesmen seem to direct and rule by a sort of power to put themselves in the place of the nation over which they are set, and may thus be said to possess the souls of poets at the same time they display the coarser sense and the more vulgar sagacity of practical men of business.”
—Woodrow Wilson (1856–1924)