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Difference between revisions of "Additivity of integral"
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Latest revision as of 15:10, 13 October 2012
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,
$$\displaystyle\int_{Q \cup R} f(u) dV = \displaystyle\int_Q f(u) dV + \displaystyle\int_R f(u) dV - \displaystyle\int_{Q \cap R} f(u) dV.$$