Question

The area of the region bounded by the curves $y=|x-2|,x=1,x=3$ and the x-axis is

Answer 1

Required area $={\int }_1^{3}ydx={\int }_1^3|x-2|dx$

$={\int }_1^2|x-2|dx={\int }_2^3|x-2|dx={\int }_1^2(2-x)dx={\int }_2^3(x-2)dx$

$={|2x-\frac{x^2}{2}|}_1^2+|\frac{x^2}{2}-2x|$

$=(4-2)-(2-\frac{1}{2})+[(\frac{9}{2}-6)-(2-4)]$

$=2-\frac{3}{3}-\frac{3}{2}+2=4-3=1$