# Riemann multiple integral

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

Let $f:{\mathbb{R}}^{n}\to \mathbb{R}$ 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

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

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

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

$$\int f(x)\mathit{d}x:={S}^{*}(f)={S}_{*}(f).$$ |

Clearly one has ${S}^{*}(f,P)\ge {S}_{*}(f,P)$. Also one has ${S}^{*}(f,P)\ge {S}_{*}(f,{P}^{\prime})$ 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 ${S}^{*}(f,P)\ge {S}^{*}(f,{P}^{\prime \prime})\ge {S}_{*}(f,{P}^{\prime \prime})\ge {S}_{*}(f,{P}^{\prime})$. So, to prove that a function is Riemann-integrable it is enough to prove that for every $\u03f5>0$ there exists a polyrectangle $P$ such that $$.

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^{}

$${\chi}_{D}(x):=\{\begin{array}{cc}1\hfill & \text{if}x\in D\hfill \\ 0\hfill & \text{otherwise}\hfill \end{array}$$ |

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

$$\mathrm{\mathbf{m}\mathbf{e}\mathbf{a}\mathbf{s}}(D):=\int {\chi}_{D}(x)\mathit{d}x.$$ |

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 $f:D\to \mathbb{R}$ be a bounded function. We say that $f$ is *Riemann measurable* if the function $\overline{f}:{\mathbb{R}}^{n}\to \mathbb{R}$

$$\overline{f}(x):=\{\begin{array}{cc}f(x)\hfill & \text{if}x\in D\hfill \\ 0\hfill & \text{otherwise}\hfill \end{array}$$ |

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

$${\int}_{D}f(x)\mathit{d}x:=\int \overline{f}(x)\mathit{d}x$$ |

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

Title | Riemann multiple integral |

Canonical name | RiemannMultipleIntegral |

Date of creation | 2013-03-22 15:03:34 |

Last modified on | 2013-03-22 15:03:34 |

Owner | paolini (1187) |

Last modified by | paolini (1187) |

Numerical id | 14 |

Author | paolini (1187) |

Entry type | Definition |

Classification | msc 26A42 |

Related topic | Polyrectangle |

Related topic | RiemannIntegral |

Related topic | Integral2 |

Related topic | AreaOfPlaneRegion |

Related topic | DevelopableSurface |

Related topic | VolumeAsIntegral |

Related topic | AreaOfPolygon |

Related topic | MoscowMathematicalPapyrus |

Related topic | IntegralOverPlaneRegion |

Defines | Riemann integrable |

Defines | Peano Jordan |

Defines | measurable |

Defines | area |

Defines | volume |

Defines | Jordan content |