Using the method of integration find the area bounded by the curve `|x|+|y|" "=" "1` . [Hint: The required region is bounded by lines `x" "+" "y" "=" "1," "x " "y" "=" "1," "" "x" "+" "y" "=" "1` and `" "x" "" "y" "=" "1]dot`

Explanation

Given curve, `|x| +|y|=1`

In first quadrant (x gt 0, y gt 0) then line AB is
`x+y=1.`
In second quadrant (x lt 0, y gt 0) then line BC is
`-x+y=1.`
In third quadrant (x lt 0, y lt 0) then line CD is
`-x-y=1.`
In fourth quadrant (x gt 0, y lt 0) then line DA is
`x-y=1`.
Since ABCD is a square
therefore, required area = 4 `xx` (area of shaded region in first quadrant)
`=4int_(0)^(1)(1-x)dx`
`=4[x-(x^(2))/(2)]_(0)^(1)=4[(1-(1)/(2))-0]`
`=2` sq. units.