Question

Sketch the graph $y=|x-1|$. Evaluate ${\int }_{-2}^4|x-1|dx$ . What does this value of the integral represent on the graph?

Answer 1

$y=x-1$

$y=|x-1|$

${\int }_{-2}^4|x-1|={\int }_{-2}^1|x-1+{\int }_1^4|x-1|$

$|x-1|$ for $x<11-x$

$|x-1|$ for $x>1x-1$

${\int }_{-2}^1(1-x)dx+{\int }_1^4(x-1)dx$

${[x-\frac{1}{2}]}_{-2}^1+{[\frac{x^2}{2}-4]}_1^4$

$[(1-\frac{1}{2})-(-2-\frac{4}{2})]+[\left(\frac{4^2}{2}\right)-(\frac{1}{2})-1]$

$[\frac{1}{2}+4]+[4+\frac{1}{2}]=8+1=9$ sq.units.