# Integral of a Sum

• March 15th 2009, 11:13 PM
manjohn12
Integral of a Sum
Theorem. Let $a,b,c \in \mathbb{R}$. Let $I$ be the smallest closed interval containing $a,b,$ and $c$ and suppose that $I \subseteq K \subseteq \mathbb{R}$. If $f: K \to \mathbb{R}$ is a function, then the integral $\smallint_{a}^{b} f$ exists if and only if the integrals $\smallint_{a}^{c} f$ and $\smallint_{b}^{c} f$ exist. So $\smallint_{a}^{b} f = \smallint_{a}^{c} f + \smallint_{c}^{b} f$.

Outline of Proof. (1) We want to show that $f$ is Riemann integrable on $[a,c]$ and $[c,b]$. Let $\epsilon >0$. Try to find a corresponding $\delta$. Let $P_1$ and $P_2$ be partitions of $[a,c]$ and $[c,b]$ respectively with meshes less than $\delta$. Then look at $|\mathcal{R}(f,P_1)-\mathcal{R}_{2}(f,P_2)|$. (2) Show that any Riemann sum of $f$ on $P$ is within $\epsilon$ of $\smallint_{a}^{c} f + \smallint_{c}^{b} f$. We know that $f$ is bounded on $[a,b]$. Suppose $|f| \leq M$. Find a $\delta_1$ corresponding to $\epsilon/3$ on $[a,c]$ and $\delta_2$ corresponding to $\epsilon/3$ on $[c,b]$. Choose $\delta < \min(\delta_1, \delta_2, \eta)$ where $\eta$ is a function of $\epsilon$ and $M$. Now consider the following:

$\left|\mathcal{R}(f,P)- \left(\int_{a}^{c} f + \int_{c}^{b} f \right) \right|$

Here we have to be careful about the endpoints?

Is this in general correct? Any other faster ways of proving this?