We have to find the area enclosed by the parabola whose equation is y 2 =4x, and the line y=2x. Draw the graphs of the equations and shade the common region.
![](https://search-static.byjusweb.com/question-images/scholr/scholr-acadecraft/Images/HTML_54803_1.JPG)
Figure (1)
The area of the region OBAO is,
Area of the region OBAO=Area of the region OBACO−Area of the triangle OAC
To find the area bounded by the straight line y=2x with the x-axis, assume a vertical strip of infinitely small strip and integrate the area.
Area of the triangle OAC= ∫ 0 1 2xdx = [ 2 x 2 2 ] 0 1 =[ 1−0 ] =1 sq units
Similarly, find the area bounded by the parabola with x-axis,
Area of the region OBACO= ∫ 0 1 ydx = ∫ 0 1 2 x dx
Simplify further,
Area of the region OBACO=2 [ x 1 2 +1 1 2 +1 ] 0 1 =2⋅ 2 3 [ x 3 2 ] 0 1 = 4 3 [ 1 ] = 4 3 sq units
Area of the required region is,
Area of the region OBAO=Area of the region OBACO−Area of the triangle OAC = 4 3 −1 = 1 3
The area of the shaded region is 1 3 sq units.
Thus, out of all the four options, option (B) is correct.