$\int\limits_0^4 {\left| {x - 1} \right|dx}$

Let $I = \int\limits_0^4 {\left| {x - 1} \right|dx}$

$\left| {x - 1} \right| = \left\{ \begin{array}{l} - \left( {x - 1} \right),\,\,\,\,if\,x < 1\\x - 1,\,\,\,\,\,\,\,\,\,\,\,if\,x \ge 1\end{array} \right.$

$\therefore$ $I = - \int\limits_0^1 {\left( {x - 1} \right)dx} + \int\limits_1^4 {\left( {x - 1} \right)dx} = - \left[ {\cfrac{{{{\left( {x - 1} \right)}^2}}}{2}} \right]_0^1 + \left[ {\cfrac{{{{\left( {x - 1} \right)}^2}}}{2}} \right]_1^4$

$= - \cfrac{1}{2}\left[ {0 - 1} \right] + \cfrac{1}{2}\left[ {9 - 0} \right] = \cfrac{1}{2} + \cfrac{9}{2} = 5$