Additivity of integral

Very familiar: $$\int_a^b f(x) dx + \int_b^c f(x) dx = \int_a^c f(x) dx.$$

This is where it comes from...

Let $f: {\bf R}^n \rightarrow {\bf R}$, then we define the Riemann integral as

$$\displaystyle\int_Q f(n) dV = \displaystyle\lim_{m \rightarrow \infty} \displaystyle\sum_i f( e_i ) \Delta V,$$

an $(n+1)$-dimensional volume, where $\Delta V$ an $n$-dimensional volume of an $n$-dimensional box.

Main property:

Integration is additive with respect to the union of domains of integration.

Suppose $Q, R$, with $Q \cap R = \emptyset$. Then

$$\displaystyle\int_{Q \cup R} f(u) dV = \displaystyle\int_Q f(u) dV + \displaystyle\int_R f(u) dV$$

What if $Q \cap R \neq \emptyset$? In this case,

$$\displaystyle\int_{Q \cup R} f(u) dV = \displaystyle\int_Q f(u) dV + \displaystyle\int_R f(u) dV - \displaystyle\int_{Q \cap R} f(u) dV.$$