Mathematics

Evaluate the integral $$\displaystyle\int_{-4}^{4}|x+2|\ dx$$.


SOLUTION
First find where 
$$x+2>0\\ x>-2$$

Thus the value of  the function in $$\left[ -4,-2 \right]$$ is $$\left| x+2 \right| =-\left( x+2 \right) $$
and  the value of function in the interval $$ \left[ -2,4 \right] $$ is $$\left| x+2 \right| =x+2$$.

Hence the integral can be  written as,
$$\int _{ -4 }^{ 4 }{ \left| x+2 \right| dx } =\int _{ -4 }^{ -2 }{ -\left( x+2 \right)  } dx+\int _{ -2 }^{ 4 }{ \left( x+2 \right) dx } \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad =-{ \left[ \frac { { x }^{ 2 } }{ 2 } +2x \right]  }_{ -4 }^{ -2 }+{ \left[ \frac { { x }^{ 2 } }{ 2 } +2x \right]  }_{ -2 }^{ 4 }\\ \quad \quad \quad \quad \quad \quad \quad \quad \quad =-\left[ 2-4-(8-8) \right] +[8+8-(2-4)]\\ \quad \quad \quad \quad \quad \quad \quad \quad \quad =2+16+2\\ \quad \quad \quad \quad \quad \quad \quad \quad \quad =20$$
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Subjective Medium Published on 17th 09, 2020
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