Question

Using the method of integration find the area bounded by the curve $|x|+|y|=1$

Answer 1

The required region is bounded by lines $x+y=1,x-y=1,-x+y=1$ and $-x-y=11$
The area bounded by the curve $|x|+|y|=1$ is represented by the shaded region $ADCB$
The curve intersects the axes at points $A(0,1),B(1,0),C(0,-1)$ and $D(-1,0)$.
It can be observed that the given curve is symmetrical about $x$-axis and $y$-axis.
$\therefore AreaADCB=4\times AreaOBAO$
$=4{\int }_0^1(1-x)dx$
$=4{(x-\frac{x^2}{2})}_0^1$
$=4[1-\frac{1}{2}]$
$=4\left(\frac{1}{2}\right)$
$=2$ sq. units