Find by integration the area of the region bounded by the curve `y=2x-x^2` and the x-axis.

Options

A. `(2)/(3)` sq units
B. `(4)/(3)` sq units
C. `(5)/(3)` sq units
D. `(8)/(3)` sq units

Answer

B

Explanation

Given equation of curves is `y=2x-x^(2)`
`Rightarrow x^(2)-2x=-y`
`Rightarrow x^(2)-2x+1=-y+1`
`Rightarrow (x-1)^(2)=-(y-1)`
This is the equation of parabola having vertex (1,1) and open downward.

The parabola intersect the X-axis put y=0, we get
`0=2x-x^(2)`
`Rightarrow x(2-x)=0`
`Rightarrow x=0,2`
`therefore` Area of bounded region between the curve and X-axis.
`=underset(0)overset(2)int (2x-x^(2))dx=[(2x^(2))/(2)-(x^(3))/(3)]_(0)^(2)`
`=[4-(8)/(3)-0=0]=(4)/(3)` sq units.