---
tag: FE1ZZJ8QW2
kind: concept
name: Multiple integral
msc: [26B15]
sources:
  - url: https://en.wikipedia.org/wiki/Multiple_integral
same_as:
  wikidata: Q844937
  wikipedia: Multiple integral
---

## Summary

A multiple integral is the integral of a real function of $n$ real variables over a region of $\R^n$.
It extends the one-variable [[Riemann integral|0W29TBTY12]] from intervals to boxes and to more general bounded sets.
The [[double integral|0WAR4F23B4]] and the [[triple integral|0698M7C7AX]] are the cases $n = 2$ and $n = 3$.
Multiple integrals give areas, volumes, masses, centres of mass and probabilities of random vectors.

## Explanations

### L1 human

Cut a region of space into many small boxes.
In each box, multiply a value of the function by the volume of the box, and add up all these products.
When the boxes get smaller and the sums settle to one number, that number is the multiple integral.
For example, the integral of the constant $1$ over the unit cube $[0, 1]^3$ is its volume, $1$.
In practice one often computes a multiple integral as an [[iterated integral|0WZCHE1ZF9]], one variable at a time.