Question

Evaluate the following integrals:

$\underset{-4}{\overset{4}{\int }}\left|x+2\right|dx$

Answer 1

$$\begin{gathered} {\int }_{-4}^4\left|x+2\right|\mathit{d}x \\ \mathrm{We}\mathrm{know}\mathrm{that},\left|x+2\right|=\left\{\begin{array}{l}-\left(x+2\right),-4\le x\le -2\\ x+2,-2<x\le 4\end{array}\right. \\ \therefore I={\int }_{-4}^4\left|\mathrm{x}+2\right|\mathit{d}x \\ \Rightarrow I={\int }_{-4}^{-2}-\left(x+2\right)dx+{\int }_{-2}^4\left(x+2\right)\mathit{d}x \\ \Rightarrow I={\left[-\frac{x^2}{2}-2x\right]}_{-4}^{-2}+{\left[\frac{x^2}{2}+2x\right]}_{-2}^4 \\ \Rightarrow I=-2+4-8-8+8+8-2+4 \\ \Rightarrow I=20 \end{gathered}$$