Question

Area of the region bounded by $y=|x-1|$ and $y=1$ is

• A. $2$ sq. units
• B. $1$ sq. units
• C. $\frac{1}{2}$ sq. units
• D. none of these

Answer 1

Answer: (B)

We have, $y=x-1$, if $x-1\ge 0$
$y=-x+1$ , if $x-1<0$

Required area = area of shaded region
$A=\underset{0}{\overset{2}{\int }}1dx-\left[\underset{0}{\overset{1}{\int }}\left(1-x\right)dx+\underset{1}{\overset{2}{\int }}\left(x-1\right)dx\right]$

$={\left[x\right]}_0^2-{\left[x-\frac{x^2}{2}\right]}_0^1-{\left[\frac{x^2}{2}-x\right]}_1^2$

$=2-\frac{1}{2}-\frac{1}{2}=1$ sq. unit