Question

The area bounded by the curve $|x+y|+|x-y|=1$ is

• A. $\frac{1}{2}$ sq. units
• B. $1$ sq. units
• C. $2$ sq. units
• D. $4$ sq. units

Answer 1

Answer: (B)

We have, $\left|x+y\right|+\left|x-y\right|=1$
$\left(i\right)$ If $x,y\ge 0$
$\left(a\right)$ If $x<y$
$x+y+x-y=1$
$\Rightarrow x=\frac{1}{2}$
$\left(b\right)$ If $x>y$
$x+y+y-x=1$
$\Rightarrow y=\frac{1}{2}$
$\left(ii\right)$ If $x,y<0$
$\left(a\right)$ If $x<y$
$-\left(x+y\right)-\left(x-y\right)=1$
$\Rightarrow x=-\frac{1}{2}$
$\left(b\right)$ If $x>y$
$-\left(x+y\right)-\left(y-x\right)=1$
$\Rightarrow y=-\frac{1}{2}$

Required area $=4\underset{0}{\overset{1/2}{\int }}\frac{1}{2}dx=2{\left[x\right]}_0^{1/2}$
$=1$ sq. unit