Question

Sketch the graph of $f\left(x\right)=\{\begin{array}{cc}|x-1|+2,& x\le 2\\ x^2-2,& x>2\end{array}$. Evaluate ${\int }_0^{4}f\left(x\right)dx$. What does the value of this integral represent on the graph?

Answer 1

For $x<1f\left(x\right)=1-x+2$ i.e $f\left(x\right)=3-x$

for $2\ge x\ge 1f\left(x\right)=x+1$

for $x>1f\left(x\right)=x^2-2$

Intefration angles to the area under the curve.

$\therefore {\int }_0^{4}f\left(x\right)$

${\int }_0^{1}3-x+{\int }_1^{2}x+1+{\int }_4^2{x}^2-2$

$=59/3$