Sum rule 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.

Integration preserves addition.

Let $f, g$ continuous. Then

$$\begin{array}{} \int_Q ( f(u) + g(u) ) dV &= \lim_{i \rightarrow \infty} \sum_i ( f(c_i) + g(c_i) ) \Delta V \\ &= \lim_{i \rightarrow \infty}( \sum_i f(c_i) \Delta V + \sum_i g(c_i) \Delta V ) \\ &= \lim_{i \rightarrow \infty} \sum_i f(c_i) \Delta V + \lim_{i \rightarrow \infty} \sum_i g(c_i) \Delta V {\rm \hspace{3pt} (sum \hspace{3pt} rule \hspace{3pt} of \hspace{3pt} limits)} \\ &= \int_Q f(u) dV + \int_Q g(u) dV. \end{array}$$

Note: Even though this property is about addition, there is another additivity property, additivity of integral.