#### Chapter 8 Integral Calculus

Section 8.2 Definite Integral

# 8.2.3 Calculation Rules

##### Partition of the Interval of Integration 8.2.5
Let $f:\left[a;b\right]\to ℝ$ be an integrable function. Then for every number $z$ between $a$ and $b$, we have

${\int }_{a}^{b}f\left(x\right) dx={\int }_{a}^{z}f\left(x\right) dx+{\int }_{z}^{b}f\left(x\right) dx .$

With the definition

${\int }_{d}^{c}f\left(x\right) dx:=-{\int }_{c}^{d}f\left(x\right) dx$

the rule above applies to all real numbers $z$ for which the two integrals on the right-hand side of the equation exist, even if $z$ does not lie between $a$ and $b$. Before we demonstrate this calculation with an example, we will examine the definition above in more detail.
##### Exchanging the Limits of Integration 8.2.6
Let $f:\left[a;b\right]\to ℝ$ be an integrable function. The integral of the function $f$ from $a$ to $b$ is calculated according to the rule

${\int }_{b}^{a}f\left(x\right) dx=-{\int }_{a}^{b}f\left(x\right) dx .$

The calculation rule described above is convenient when integrating functions that involve absolute values, or piecewise-defined functions.
##### Example 8.2.7
The integral of the function $f:\left[-4;6\right]\to ℝ,x↦|x|$ is

$\begin{array}{ccc}\multicolumn{1}{c}{{\int }_{-4}^{6}|x| dx}& =\hfill & {\int }_{-4}^{0}\left(-x\right) dx+{\int }_{0}^{6}x dx\hfill \\ \multicolumn{1}{c}{}& =\hfill & {\left[-\frac{1}{2}{x}^{2}\right]}_{-4}^{0}+{\left[\frac{1}{2}{x}^{2}\right]}_{0}^{6}\hfill \\ \multicolumn{1}{c}{}& =\hfill & \left(0-\left(-8\right)\right)+\left(18-0\right)\hfill \\ \multicolumn{1}{c}{}& =\hfill & 26 .\hfill \end{array}$

The integration over a sum of two functions can also be split up into two integrals:
##### Sum and Constant Multiple Rule 8.2.8
Let $f$ and $g$ be integrable functions on $\left[a;b\right]$, and let $r$ be a real number. Then

${\int }_{a}^{b}\left(f\left(x\right)+g\left(x\right)\right) dx={\int }_{a}^{b}f\left(x\right) dx+{\int }_{a}^{b}g\left(x\right) dx .$ $\left(8.2.2\right)$

For constant multiples of a function, we have

${\int }_{a}^{b}r·f\left(x\right) dx=r·{\int }_{a}^{b}f\left(x\right) dx .$ $\left(8.2.3\right)$