What is the area bounded by the curve `y = 4x-x^(2) -3` and the x-axis ?

Options

A. `2//3` sq units.
B. `4//3` sq unit
C. `5//3` sq unit
D. `4//5` sq unit

Answer

B

Explanation

Given curve is `y = 4x - x^(3) - 3`
Since, area bounded by x-axis
`:. y = 0`
`rArr 4x -x^(2) - 3 = 0 rArr x^(2) - 4x + 3 = 0`
`rArr x^(2) - 3x -x + 3 = 0 rArr (x-3)(x-1) = 0`
`rArr x = 1,3`
`:.` Required area `= int_(1)^(3) (4x-x^(2)-3)dx`
`= (4x^(2))/(2) - (x^(3))/(3) - 3x|_(1)^(3) = ((36)/(2) - (27)/(3) - 9) - (4/2 - 1/3 - 3)`
`= (18- 9-9) - (2-(10)/(3)) = 0- ((- 4)/(3 ) ) = 4/3`sq. unit