Properties of Definite Integrals
Throughout, we assume $f$ and $g$ are arbitrary functions and $c$ is any constant:

$\int_a^b c \; dx = c(ba)$.
 $\int_a^b [f(x) \pm g(x)] \,dx = \int_a^b f(x) dx \pm \int_a^b g(x) \,dx$
 $\int_a^b cf(x) \,dx = c\int_a^b f(x) \,dx$
 $\int_b^a f(x)\, dx = \int_a^b f(x) \,dx$
 $\int_a^a f(x) \,dx = 0$

If one function is bigger than another, then its definite integral
from $a$ to $b$ (with $a \le b$) will
also be bigger. In particular,
 If $a \le b$ and $f(x) \geq 0$ for $a \leq x \leq b$, then
$\int_a^b f(x) dx \geq 0$.
 If $a \le b$ and $f(x) \geq g(x)$ for $a \leq x \leq b$, then
$\int_a^b f(x)\, dx \geq \int_a^b g(x)\, dx$.
 If $a \le b$ and $m \leq f(x) \leq M$ for $a \leq x \leq b$,
then $$m(ba) \leq \int_a^b f(x)\, dx \leq M(ba)$$ (Why?)

In other words, we can compare $f$ to $0$, we can compare $f$ to $g$,
and we can compare $f$ to $m$ and $M$.
These properties are easy to visualize if you think about area. If
$f(x) \ge g(x)$, then the curve $y=f(x)$ lies above the curve $y=g(x)$,
so there is more area under the $f$ curve than under the $g$ curve.
But these properties apply even when we are dealing with negative functions
or applications that have nothing to do with area. For instance, if
$f(x) \ge g(x)$ for all $x \in [a,b]$, then for each $i=1,\ldots,n$
$$
f(x_i^*) \ge g(x_i^*)
\quad \Longrightarrow
\quad f(x_i^*)\,\Delta x \ge g(x_i^*)\,\Delta x. $$
Hence,
$$
{\sum_{i=1}^n f(x_i^*)\, \Delta x} \ge
{\sum_{i=1}^n g(x_i^*)\, \Delta x}. $$ By limit laws, $$
{\lim_{n \to \infty}\, \sum_{i=1}^n f(x_i^*)\, \Delta x} \ge
{\lim_{n \to \infty}\,\sum_{i=1}^n g(x_i^*)\, \Delta x},
$$
and by definition of integral
$$\int_a^b f(x)\, dx \ge \int_a^b g(x)\, dx.
$$
