What is the area of the region bounded by theparabolas `y^2 = 6 (x - 1) and y^2= 3x ?`

Options

A. `(sqrt(6))/(3)`
B. `(2sqrt(6))/(3)`
C. `(4sqrt(6))/(3)`
D. `(5sqrt(6))/(3)`

Answer

C

Explanation

Given parabolas are `y^(2) = 6(x-1)"……."(1)`
and `y^(2) = 3x "……."(2)`

`y^(2) = 6(x-1)`
`rArr 6x-6 = y^(2)`
`rArr x = (y^(2) + 6)/(6) = (y^(2))/(6) + 1`
Also, `y^(2) = 3x rArr x = (y^(2))/(3)`.
Sloving `(1), (2), 3x = 6(x-1)`
`rArr 3x = 6x-6`
`rArr 3x = 6rArr x = 2`.
`y^(2) = 6 rArr y = +-sqrt(6)`.
`:.` Area `=`
`int_(-sqrt(6))^(sqrt(6))((y^(2))/(6) + 1 - (y^(2))/(3)).dy = 2int_(0)^(sqrt(6))(1-(y^(2))/(6)) dy = 2[y - (y^(3))/(18)]_(0)^(sqrt(6))`
`= 2 [sqrt(6) - ((sqrt(6))^(3))/(18)] = 2 xx (2sqrt(6))/(3) = (4sqrt(6))/(3)`.