#### Definition 10.1.1.

Let \((a_1,a_2,\ldots,a_n)\) and \((b_1,b_2,\ldots,b_n)\) be such that \(a_k \leq b_k\) for all \(k\text{.}\) The set \([a_1,b_1] \times
[a_2,b_2] \times \cdots \times
[a_n,b_n]\) is called a

*closed rectangle*. In this setting it is sometimes useful to allow \(a_k = b_k\text{,}\) in which case we think of \([a_k,b_k] = \{ a_k \}\) as usual. If \(a_k < b_k\) for all \(k\text{,}\) then the set \((a_1,b_1) \times (a_2,b_2) \times \cdots \times (a_n,b_n)\) is called an*open rectangle*.For an open or closed rectangle \(R \coloneqq [a_1,b_1] \times
[a_2,b_2] \times \cdots \times
[a_n,b_n] \subset \R^n\) or \(R \coloneqq (a_1,b_1) \times
(a_2,b_2) \times \cdots \times
(a_n,b_n) \subset \R^n\text{,}\) we define the

*\(n\)-dimensional volume*by
\begin{equation*}
V(R) \coloneqq
(b_1-a_1)
(b_2-a_2)
\cdots
(b_n-a_n) .
\end{equation*}

A

*partition*\(P\) of the closed rectangle \(R = [a_1,b_1] \times [a_2,b_2] \times \cdots \times [a_n,b_n]\) is given by partitions \(P_1,P_2,\ldots,P_n\) of the intervals \([a_1,b_1], [a_2,b_2],\ldots, [a_n,b_n]\text{.}\) We write \(P=(P_1,P_2,\ldots,P_n)\text{.}\) That is, for every \(k=1,2,\ldots,n\) there is an integer \(\ell_k\) and a finite set of numbers \(P_k = \{ x_{k,0},x_{k,1},x_{k,2},\ldots,x_{k,\ell_k} \}\) such that
\begin{equation*}
a_k = x_{k,0} < x_{k,1} < x_{k,2} < \cdots < x_{k,{\ell_k}-1} < x_{k,\ell_k} = b_k .
\end{equation*}

Picking a set of \(n\) integers \(j_1,j_2,\ldots,j_n\) where \(j_k \in \{ 1,2,\ldots,\ell_k \}\) we get the

*subrectangle*
\begin{equation*}
[x_{1,j_1-1}~,~ x_{1,j_1}]
\times
[x_{2,j_2-1}~,~ x_{2,j_2}]
\times
\cdots
\times
[x_{n,j_n-1}~,~ x_{n,j_n}] .
\end{equation*}

We order the subrectangles somehow and we say \(\{R_1,R_2,\ldots,R_N\}\) are the subrectangles corresponding to the partition \(P\) of \(R\text{,}\) or more simply, subrectangles of \(P\text{.}\) In other words, we subdivided the original rectangle into many smaller subrectangles. See Figure 10.1.

Let \(R \subset \R^n\) be a closed rectangle and let \(f \colon R \to \R\) be a bounded function. Let \(P\) be a partition of \(R\) with \(N\) subrectangles \(R_1,R_2,\ldots,R_N\text{.}\) Define

\begin{equation*}
\begin{aligned}
& m_i \coloneqq \inf \bigl\{ f(x) : x \in R_i \bigr\} , &
& M_i \coloneqq \sup \bigl\{ f(x) : x \in R_i \bigr\} , \\
& L(P,f) \coloneqq
\sum_{i=1}^N m_i V(R_i) , &
& U(P,f) \coloneqq
\sum_{i=1}^N M_i V(R_i) .
\end{aligned}
\end{equation*}

We call \(L(P,f)\) the

*lower Darboux sum*and \(U(P,f)\) the*upper Darboux sum*.