Answer
12.5
Work Step by Step
The function $f(x)=|2x-5|=\left\{\begin{array}{lll}
2x-5, & when & 2x-5 \geq 0\\
-(2x-5) & when & 2x-5 < 0
\end{array}\right.$
$f(x)=\left\{\begin{array}{lll}
2x-5, & when & x \geq 5/2\\
-(2x-5) & when & x < 5/2
\end{array}\right.$
Apply Th.4.6 from section 4-3, Additive Interval Property
$\displaystyle \int_{0}^{5}|2x-5|dx=\int_{0}^{5/2}(5-2x)dx+\int_{5/2}^{5}(2x-5)dx$
Use the table "Basic Integration Rules", p.246
$=[5x-x^{2}]_{0}^{5/2}+[x^{2}-5x]_{5/2}^{5}$
$=(\displaystyle \frac{25}{2}-\frac{25}{4})-0+(25-25)-(\frac{25}{4}-\frac{25}{2})$
$=2(\displaystyle \frac{25}{2}-\frac{25}{4})$
$=\displaystyle \frac{25}{2}$
Checking graphically - see below (work done in desmos.com).
Sum of the areas of two right triangles,
each with legs $2.5$ and 5,
$A=2\displaystyle \cdot[\frac{1}{2}(2.5)(5))=12.5$
