Find the area bounded by the curve `y=2x-x^2` and the straight line `y=-x`

Options

A. `(13)/(2)` sq unit
B. `(9)/(2)` sq unit
C. `(7)/(2)` sq unit
D. `(21)/(2)` sq unit

Answer

B

Explanation

Intersection points of the given curves are (0, 0) and (3, 3).
`therefore` Required area `=int_(0)^(3)(y_(1)-y_(2))dx`
`=int_(0)^(3){2x-x^(2)-(-x)}dx`
`[{:(becauseP(x,y_(1))andQ(x,y_(2))" lie on "y=2x-x^(2)),("and y = - x respectively."),(therefore y_(1)=2x-x^(2)andy_(2)=-x):}]`

`=int_(0)^(3)(3x-x^(2))dx`
`=[(3)/(2)x^(2)-(x^(3))/(3)]_(0)^(3)=(27)/(2)-(27)/(3)=(9)/(2)` sq units