Question

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

Answer 1

Given equation of curve is $|x|+|y|=1$.

$⟹\pm x\pm y=1$

The above equation, represents these four lines,

$x+y=1$

$x-y=1$

$-x+y=1$

$-x-y=1$

The graphical representation of these lines is shown in figure.

Since, the required area is symmetrical in all the four quadrants.

Therefore,

Required area = 2( Area of region OABO )

$=4\underset{0}{\overset{1}{\int }}ydx=4\underset{0}{\overset{1}{\int }}(1-x)dx$

$=4\left[\right(1-\frac{1^2}{2})-(0-\frac{0^2}{2}\left)\right]$

$=4\times \frac{1}{2}=2sq.units$