The area (in square units) of the region enclosed between the parabola y^2=2 x and the line y=4 x-1

Step-by-step solution

Given equation of parabola y^2=2 x ( i ) and equation of line y=4 x-1 ( ii ) Intersecting points of Eqs. (i) and (ii) are ( 1 8 , -1 2 ) and ( 1 2 , 1 ) Required area A O B= _ -1 / 2 ^1 ( y+1 4 - y^2 2 ) d y= 1 4 _ -1 / 2 ^1(y+1) d y- 1 2 _ -1 / 2 ^1 y^2 d y= 1 4 [ y^2 2 +y ]_ -1 / 2 ^1- 1 2 [ y^3 3 ]_ -1 / 2 ^1= 1 4 [ ( 1 2 +1 )- ( 1 8 - 1 2 ) ]- 1 6 [1- (- 1 8 ) ]= 1 4 [ 3 2 + 3 8 ]- 1 6 ( 9 8 )= 1 4 15 8 - 3 16 = 15-6 32 = 9 32 sq units