Suppose that `F(alpha)` denotes the area of the region bounded by `x = 0, x=2, y^2=4x` and `y=|alphax-1|+|alphax-2|+alphax`, where `alpha in {0,1}`. Then the value of `F(alpha)+(8sqrt(2))/3` when `alpha = 0` is (A) 4 (B) 5 (C) 6 (D) 9

Options

A. 5
B. 6
C. 7
D. 9

Answer

a

Explanation

When `alpha=1` F(1) is the area of the shaded region bounded by `x=0, x=2 , y^(2)=4x and y=|x-1|+|x-2|+x`
or, `y={{:(-x+3","x lt1),(x+1","1 le x lt 2),(3x-3","x ge2):}`
as shown in Fig. 31
`therefore F(1)=underset(0)overset(1)int(y_(2)-y_(1))dx + underset(1)overset(2)int (y_(4)-y_(3))dx`

`implies F(1)=underset(0)overset(1)int (-x+3-2sqrt(x))dx + underset(1)overset(2)int(x+1-2sqrt(x))dx`
`implies F(1)=[-(x^(2))/(2)+3x-(4)/(3)x^(3//2)]_(0)^(1)+[(x^(2))/(2)+x-(4)/(3)x^(3//2)]_(1)^(2)`
`implies F(1)=(-(1)/(2)3-(4)/(3))+(2+2-(8sqrt(2))/(3)-(1)/(2)-1+(4)/(3))`
`F(1)=5-(8)/(3)sqrt(2)`
`implies F(1)+(8)/(3)sqrt(2)=5`