## You are here

HomeRiemann multiple integral

## Primary tabs

# Riemann multiple integral

We are going to extend the concept of Riemann integral to functions of several variables.

Let $f\colon\mathbb{R}^{n}\to\mathbb{R}$ be a bounded function with compact support. Recalling the definitions of polyrectangle and the definitions of upper and lower Riemann sums on polyrectangles, we define

$S^{*}(f):=\inf\{S^{*}(f,P)\colon\text{$P$ is a polyrectangle, $f(x)=0$ for % every $x\in\mathbb{R}^{n}\setminus\cup P$}\},$ |

$S_{*}(f):=\sup\{S_{*}(f,P)\colon\text{$P$ is a polyrectangle, $f(x)=0$ for % every $x\in\mathbb{R}^{n}\setminus\cup P$}\}.$ |

If $S^{*}(f)=S_{*}(f)$ we say that $f$ is *Riemann-integrable* on $\mathbb{R}^{n}$ and we define the Riemann integral of $f$:

$\int f(x)\,dx:=S^{*}(f)=S_{*}(f).$ |

Clearly one has $S^{*}(f,P)\geq S_{*}(f,P)$. Also one has $S^{*}(f,P)\geq S_{*}(f,P^{{\prime}})$ when $P$ and $P^{{\prime}}$ are any two polyrectangles containing the support of $f$. In fact one can always find a common refinement $P^{{\prime\prime}}$ of both $P$ and $P^{{\prime}}$ so that $S^{*}(f,P)\geq S^{*}(f,P^{{\prime\prime}})\geq S_{*}(f,P^{{\prime\prime}})\geq S% _{*}(f,P^{{\prime}})$. So, to prove that a function is Riemann-integrable it is enough to prove that for every $\epsilon>0$ there exists a polyrectangle $P$ such that $S^{*}(f,P)-S_{*}(f,P)<\epsilon$.

Next we are going to define the integral on more general domains. As a byproduct we also define the measure of sets in $\mathbb{R}^{n}$.

Let $D\subset\mathbb{R}^{n}$ be a bounded set. We say that $D$ is *Riemann measurable* if
the characteristic function

$\chi_{D}(x):=\begin{cases}1&\text{if $x\in D$}\\ 0&\text{otherwise}\end{cases}$ |

is Riemann measurable on $\mathbb{R}^{n}$ (as defined above). Moreover we define the *Peano-Jordan measure* of $D$ as

$\mathbf{meas}(D):=\int\chi_{D}(x)\,dx.$ |

When $n=3$ the Peano Jordan measure of $D$ is called the *volume* of $D$,
and when $n=2$ the Peano Jordan measure of $D$ is called the *area* of $D$.

Let now $D\subset\mathbb{R}^{n}$ be a Riemann measurable set and let $f\colon D\to\mathbb{R}$ be a bounded function. We say that $f$ is *Riemann measurable* if the function $\bar{f}\colon\mathbb{R}^{n}\to\mathbb{R}$

$\bar{f}(x):=\begin{cases}f(x)&\text{if $x\in D$}\\ 0&\text{otherwise}\end{cases}$ |

is Riemann integrable as defined before. In this case we denote with

$\int_{D}f(x)\,dx:=\int\bar{f}(x)\,dx$ |

the *Riemann integral* of $f$ on $D$.

## Mathematics Subject Classification

26A42*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections