What \(\rm \int_3^5 \left| {x - 4} \right|dx\) equals to?

Concept:

\(\rm \mathop \smallint \nolimits_a^b f\left( x \right)dx = \;\mathop \smallint \nolimits_a^c f\left( x \right)dx + \;\mathop \smallint \nolimits_c^b f\left( x \right)dx\)

Calculation:

f(x) = \(\rm \left| {x - 4} \right|\)

\(\rm \Rightarrow \;f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} { - \left( {\rm x - 4} \right),\;\;3 \le \rm x < 4}\\ {\rm x - 4,\;\;\;\;\;\;\;\;\;4 \le\rm x \le 5} \end{array}} \right.\)

\(\rm \therefore \;f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {{4 -\rm x } ,\;\;3 \le \rm x < 4}\\ {\rm x - 4,\;\;\;4 \le\rm x \le 5} \end{array}} \right.\)

Let, I = \(\rm \int_{3}^{5} |x - 4| dx\)

or I = \(\rm \int_{3}^{4} (4 - x )dx + \int_{4}^{5} (x-4 )dx \)

= \(\rm \left[4x - \frac {x^2}{2}\right]_3^4 + \left[\frac {x^2}{2} - 4x\right]_4^5\)

= (16 - 8) - (12 - \(\frac 9 2\)) + (\(\frac {25} 2\)- 20) - (8 - 16) 

= 8 + 8 - \(\rm [ {\frac {15} {2}} + \frac {15} {2}]\)

= 16 - 15

= 1