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**manjohn12** **Theorem**. Let $\displaystyle a,b,M $ and $\displaystyle m $ be real numbers with $\displaystyle a<b $ and $\displaystyle m \leq M $. Let $\displaystyle f $ and $\displaystyle g $ be real-valued functions that are Riemann integrable on $\displaystyle [a,b] $. Then the following hold:

1. If $\displaystyle f(x) \geq 0 $ for all $\displaystyle x \in [a,b] $ then $\displaystyle \int_{a}^{b} f \geq 0 $.

2. If $\displaystyle f(x) \leq g(x) $ for all $\displaystyle x \in [a,b] $ then $\displaystyle \int_{a}^{b} f \leq \int_{a}^{b} g $.

3. If $\displaystyle m \leq f(x) \leq M $ for all $\displaystyle x \in [a,b] $, then $\displaystyle m(b-a) \leq \int_{a}^{b} f \leq M(b-a) $.