The area of the region bounded by the curves `y^(2)=8x` and y = x (in sq unit) is

Options

A. ` ( 64 ) /( 3 ) `
B. ` ( 32 ) /( 3 ) `
C. ` ( 16 ) /( 3 ) `
D. ` ( 8 ) /( 3) `

Answer

B

Explanation

Given curves,
` y^ 2 = 8 x " " `… (i)
and ` y = x" " `… (ii)

On solving Eqs ( i) and (ii) , we get
` x ^ 2 -8 x = 0 `
` x ( x- 8 ) = 0 `
` rArr x = 0 , 8 `
` and y = 0 , 8 `
` therefore ` Required area, (OPA) = ` int _ 0 ^( 8) ( sqrt( 8x ) - x ) dx `
`= [ 2sqrt 2 * ( 2 )/( 3 ) * x ^ ( 3//2 ) - ( x ^ 2 ) /( 2 ) ] _ 0 ^ ( 8 ) `
` = ( 4 sqrt 2 ) /( 3) * ( 8 ) ^( 3//2 ) - ((8) ^ 2) /( 2 ) `
` = ( 4sqrt 2 ) / ( 3) * ( 2 ) ^ 3 * 2 sqrt2 - ( 64 ) / ( 2 ) `
` = ( 16 )/ ( 3 ) xx 8 - ( 64 ) / ( 2 ) = 64 (( 2 )/ ( 3 ) - ( 1 )/ (2 )) `
` = 64 xx (( 4 - 3 )) / ( 6 ) = 64 xx ( 1 )/( 6 ) = ( 32 ) /( 3 ) `