Question

Area bounded by the lines $y=|x|-2$ and $y=1-|x-1|$ is equal to

• A. $4$ sq. units
• B. $6$ sq. units
• C. $2$ sq. units
• D. $8$ sq. units

Answer 1

Answer: (A)

We have, $y=-x-2...(i)$
$y=x-2....(ii)$
$y=2-x.....(iii)$
$y=x...(iv)$
Solving $(iii)$ and $(iv)$, we get $A(1,1)$
Solving $(i)$ and $(iv)$ we get $D(-1,-1)$
Required area = area of $\Delta AOB$ + area of $\Delta OCB$ + area of $\Delta OCD$

Area of $\Delta AOB=\underset{0}{\overset{1}{\int }}xdx+\underset{1}{\overset{2}{\int }}\left(2-x\right)dx$
$={\left[\frac{x^2}{2}\right]}_0^1+{\left[2x-\frac{x^2}{2}\right]}_1^2$
$=\frac{1}{2}+\left[4-\frac{4}{2}\right]-\left[2-\frac{1}{2}\right]$
$=\frac{1}{2}+\frac{4}{2}-\frac{3}{2}=1$ sq. units
Area of $\Delta OCB=\left|\underset{0}{\overset{2}{\int }}\left(x-2\right)dx\right|$
$=\left|{\left[\frac{x^2}{2}-2x\right]}_0^2\right|=2$ sq. units
Area of $\Delta OCD=\left|\underset{-1}{\overset{0}{\int }}\left(-x-2\right)dx-\underset{-1}{\overset{0}{\int }}xdx\right|$
$=\left|-{\left[\frac{x^2}{2}+2x\right]}_{-1}^0-{\left[\frac{x^2}{2}\right]}_{-1}^0\right|=1$ sq. unit