7. Area lying between the curves y2- 4x and y 2x is3
We have to find the area enclosed by the parabola whose equation is y 2 = 4 x , and the line y = 2 x . Draw the graphs of the equations and shade the common region.
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 = 2 x with the x-axis, assume a vertical strip of infinitely small strip and integrate the area.
Area of the triangle OAC = ∫ 0 1 2 x d x = [ 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 y d x = ∫ 0 1 2 x d x
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.
We have to find the area enclosed by the parabola whose equation is
Figure (1)
The area of the region OBAO is,
To find the area bounded by the straight line
Similarly, find the area bounded by the parabola with x-axis,
Simplify further,
Area of the required region is,
The area of the shaded region is
Thus, out of all the four options, option (B) is correct.
