Question

Evaluate :${\int }_{-2}^2\left|\left[x\right]\right|dx$

• A. $1$
• B. $2$
• C. $3$
• D. $4$
• E. $5$

Answer 1

The correct option is D

$4$

Evaluating the integral

$$\begin{gathered} I={\int }_{-2}^2\left|\left[x\right]\right|dx \\ ={\int }_{-2}^{-1}\left|\left[x\right]\right|dx+{\int }_{-1}^0\left|\left[x\right]\right|dx+{\int }_0^1\left|\left[x\right]\right|dx+{\int }_1^2\left|\left[x\right]\right|dx \\ ={\int }_{-2}^{-1}\left|-2\right|dx+{\int }_{-1}^0\left|-1\right|dx+{\int }_0^1\left|0\right|dx+{\int }_1^2\left|1\right|dx \\ =2{\int }_{-2}^{-1}dx+1{\int }_{-1}^{0}dx+{\int }_0^{1}0dx+1{\int }_1^{2}dx \\ =2{\left[x\right]}_{-2}^{-1}+1{\left[x\right]}_{-1}^0+0+1{\left[x\right]}_1^2 \\ =2\left[-1-\left(-2\right)\right]+1\left[0-\left(-1\right)\right]+\left[2-1\right] \\ =2+1+1 \\ I=4 \end{gathered}$$

Hence, option D is the correct answer.