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

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

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

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

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 $\bar{f}\colon\mathbb{R}^{n}\to\mathbb{R}$

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

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