Homogeneity of integral

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

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

where $\Delta V$ an $n$-dimensional volume of an $n$-dimensional box.

Integration preserves scalar multiplication.

$$\begin{array}{} \int_Q k f(u) dV &= \lim_{m \rightarrow \infty} \sum_i k f(c^i) \Delta V \\ &= k \lim_{m \rightarrow \infty} \sum_i f(c^i) \Delta V \\ &= k \int_Q f(u) dV \end{array}$$

See also Linearity of integral.