Properties_of_the_Integral_\(Abbott_–_Theorem_7.4.2)
Assume $f$ and $g$ are integrable functions on the interval $\left. \lbrack a,b\rbrack \right.$.
The function $f + g$ is integrable on $\left. \lbrack a,b\rbrack \right.$ with $\int_{a}^{b}\left. (f + g) \right. = \int_{a}^{b}f + \int_{a}^{b}g$.
For $k \in \mathbb{R}$, the function $kf$ is integrable with $\int_{a}^{b}kf = k\int_{a}^{b}f$.
If $m \leq f\left. (x) \right. \leq M$ on $\left. \lbrack a,b\rbrack \right.$, then $m\left. (b - a) \right. \leq \int_{a}^{b}f \leq M\left. (b - a) \right.$.
If $f\left. (x) \right. \leq g\left. (x) \right.$ on $\left. \lbrack a,b\rbrack \right.$, then $\int_{a}^{b}f \leq \int_{a}^{b}g$.
The function $|f|$ is integrable and $\left| \int_{a}^{b}f \right| \leq \int_{a}^{b}|f|$.
Proof: