Normalization of integral

Let $f: {\bf R}^n \rightarrow {\bf R}$ and

$\displaystyle\int_Q f(n) dV = \displaystyle\lim_{m \rightarrow \infty} \displaystyle\sum_i f( e_i ) \Delta V$, a $(n+1)$-dimensional volume,

where $\Delta V$ a $n$-dimensional volume of an $n$-dimensional box, is the Riemann integral.

The integral of 1 is equal to the area of the domain of integration.

$$\begin{array}{} \int_Q 1 dV &= n{\rm -dimensional \hspace{3pt} volume \hspace{3pt} of \hspace{3pt}} Q, {\rm \hspace{3pt} where \hspace{3pt}} Q \subset {\bf R}^n \\ &= \lim_{m \rightarrow \infty} \sum_i 1 \Delta V {\rm \hspace{3pt} Riemann \hspace{3pt} sum} \\ &= \lim_{m \rightarrow \infty} \sum_i \Delta V, {\rm \hspace{3pt} where \hspace{3pt}} V {\rm \hspace{3pt} are \hspace{3pt} boxes \hspace{3pt} that \hspace{3pt} add \hspace{3pt} to \hspace{3pt} the \hspace{3pt} whole \hspace{3pt}} Q &= \lim Q = Q. \end{array}.$$