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 curve is `y=2x-x^(2)`
`rArrx^(2)-2x=-y`
`rArrx^(2)-2x+1=-y+1`
`rArr(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)rArrx(2-x)=0`
`rArrx=0,2`
`therefore` Area of bounded region between the curve and X-axis `=int_(0)^(2)ydx`
`=int_(0)^(2)(2x-x^(2))dx=[(2x^(2))/(2)-(x^(3))/(3)]_(0)^(2)`
`=[4-(8)/(3)-0-0]=(4)/(3)` sq units.