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

Options

A. 1 square unit
B. `2sqrt(2)` suqare unit
C. `2`square unit
D. `2sqrt(3)` units

Answer

C

Explanation

`|x|+|y| = 1`
We know , `|x| = {(x,x ge0),(-x,x le0):}`
`:. |x| +|y| = 0`
`{{:(x+y=1 , " for " , xlt0,ylt0),(-x+y=1," for ", xlt0,ygt0),(x-y=1," for ", x gt 0, y lt 0),(-x-y=1," for " , x lt 0 , y lt 0):}`
If we plot graphs of these equations, we get

The curve is symmetrical abuot x any y-axis.
`:.` Area `= 4 xx " Area of AOD "`
`= 4 x int_(0)^(1) xx intydx = 4 xx int_(0)^(1) (1-x) dx = 4[x-(x^(2))/(2)]_(0)^(1)`
`= 4 [1-(1^(2))/(2) - (0-(0^(2))/(2))]`
`= 4 (1-1/2) = 4 xx 1/2 = 2`