## Properties of the Integral

1. Introduction. There are about 4 properties we expect the integral to obey. We expect an integral over a zero width region should vanish
(1)$\displaystyle\int^{a}_{a}f(x)\,\mathrm{d}x=0$
The fundamental theorem of calculus confirms this. If $F(x)$ is the antiderivative for $f(x)$, we see
$\displaystyle\int^{a}_{a}f(x)\,\mathrm{d}x=F(a)-F(a)=0$
identically vanishes.

Additivity. Likewise, if we have the integral be over $[a,c]$ and $a, then
(2)$\displaystyle\int^{b}_{a}f(x)\,\mathrm{d}x+\int^{c}_{b}f(x)\,\mathrm{d}x=\int^{c}_{a}f(x)\,\mathrm{d}x$
Again, using the fundamental theorem of calculus we see
$\displaystyle\int^{b}_{a}f(x)\,\mathrm{d}x+\int^{c}_{b}f(x)\,\mathrm{d}x=F(b)-F(a)+F(c)-F(b)=F(c)-F(a)$
and
$\displaystyle\int^{c}_{a}f(x)=F(c)-F(a)$
Setting equals to equals yields (2).

This should make sense, since the intuition behind integration is consider a “continuous sum” of thin rectangular strips between $y=0$ and $y=f(x)$. The width of each strip is $\mathrm{d}x$ (“infinitesimal” wink wink), and we just keep adding more and more strips. In a sense this is just the associativity of addition.

Linearity of Integrand. Let $f(x)$, $g(x)$ be continuous functions on $[a,b]$. Let $F(x)$, $G(x)$ be their antiderivatives. Then
(3)$\displaystyle\int^{b}_{a}\bigl[c_{1}f(x)+c_{2}g(x)\bigr]\,\mathrm{d}x=c_{1}\int^{b}_{a}f(x)\,\mathrm{d}x+c_{2}\int^{b}_{a}g(x)\,\mathrm{d}x$
where $c_1,c_2$ are constants.

How do we see this? Again, by the fundamental theorem of calculus we have
$\displaystyle \begin{array}{rl}\displaystyle\int^{b}_{a}\bigl[c_{1}f(x)+c_{2}g(x)\bigr]\,\mathrm{d}x &=\displaystyle\bigl(c_{1}F(b)+c_{2}G(b)\bigr)-\bigl(c_{1}F(a)+c_{2}G(a)\bigr)\\ &=\displaystyle c_{1}\bigl(F(b)-F(a)\bigr)+c_{2}\bigl(G(b)-G(a)\bigr)\end{array}$
We see this is precisely the right hand side of (3), after invoking the fundamental theorem of calculus.

Special cases of linearity: $f(x)+g(x)$ taking $c_{1}=c_{2}=1$; $f(x)-g(x)$ taking $c_{1}=1$ and $c_{2}=-1$. Also $kf(x)$ where $k$ is a constant, take $c_{1}=k$, $c_{2}=0$.

Domination Property. On an interval $[a,b]$ suppose $f(x)\geq g(x)$ for each $x\in[a,b]$. Then
(4)$\displaystyle\int^{b}_{a}f(x)\,\mathrm{d}x\geq \int^{b}_{a}g(x)\,\mathrm{d}x$
The “proof” here is more intuitive: $f(x)-g(x)\geq0$ for each $x\in[a,b]$. Thus its integral over $[a,b]$ is positive.

Inequalities. Take $f(x)$ continuous on $[a,b]$. Let $\min(f)\leq f(x)\leq\max(f)$ for each $x\in[a,b]$. Then
(5)$\displaystyle\min(f)\cdot(b-a)\leq\int^{b}_{a}f(x)\,\mathrm{d}x\leq \max(f)\cdot(b-a)$
This can follow from the domination property if we take $g(x)=\min(f)$ and $h(x)=\max(f)$ and observe $g(x)\leq f(x)\leq h(x)$ for each $x\in[a,b]$.