Question

${\int }_{-3}^3(\left|x\right|+|x-2|)dx+30=$

Answer 1

$f\left(x\right)=|x|+|x-2|\Rightarrow f\left(x\right)=\{\begin{array}{c}-x-(x-2),-3<x<0\\ x-(x-2),0\le x<2\\ x+x-2,2\le x\le 3\end{array}\Rightarrow f\left(x\right)=\{\begin{array}{c}2(1-x),-3<x<0\\ 2,0\le x<2\\ 2(x-1),2\le x\le 3\end{array}\Rightarrow {\int }_3^{-3}f\left(x\right)dx=-{\int }_{-3}^{3}f\left(x\right)dx=-\left({\int }_{-3}^{0}f\right(x)dx+{\int }_0^{2}f(x)dx+{\int }_2^{3}f(x\left)dx\right)=-\left(\right(2x-x^2{)}_{-3}^0+(2x{)}_0^2+(x^2-2x{)}_2^3)=-(-6-9+4+9-6-4+4)=8$