Question

Evaluate the following integrals:
$\underset{0}{\overset{4}{\int }}\left|x-1\right|dx$

Answer 1

$$\begin{gathered} {\int }_0^4\left|x-1\right|\mathit{d}x \\ \mathrm{We}\mathrm{know}\mathrm{that},\left|x-1\right|=\left\{\begin{array}{l}-\left(x-1\right),0\le x\le 1\\ x-1,1<x\le 4\end{array}\right. \\ \therefore I={\int }_0^4\left|x-1\right|\mathit{d}x \\ \Rightarrow I={\int }_0^1-\left(x-1\right)dx+{\int }_1^4\left(x-1\right)dx \\ \Rightarrow I={\left[-\frac{x^2}{2}+x\right]}_0^1+{\left[\frac{x^2}{2}-x\right]}_1^4 \\ \Rightarrow I=\frac{-1}{2}+1-0+8-4-\frac{1}{2}+1 \\ \Rightarrow I=5 \end{gathered}$$