Question

Evaluate ${\int }_0^4(|x|+|x-2|+|x-4|)$dx.

Answer 1

${\int }_0^4(\left|x\right|+|x-2|+|x-4|)dx$

${\int }_0^4\left|x\right|+{\int }_0^{2}2-x+{\int }_0^{4}x-2+{\int }_0^{4}4-x$

${}_0^4|\frac{x^2}{2}+{[2x-\frac{x^2}{2}]}_0^2+{[\frac{x^2}{2}-2x]}_2^4+{[4x-\frac{x^2}{2}]}_0^4$

$=8+4-2+8-8-2+4+16-8$

$=32-12$

$=20$