Draw a rough sketch and find the area of the region bounded by the two parabolas y2 = 4x and x2 = 4y by using methods of integration.
To find the area bounded by
y2 = 4x
And x2 = 4y
On solving the equation (i) and (ii),
Or, x4 – 64x = 0
Or, x(x3 – 64) = 0
Or, x = 0, 4
Then y = 0,4
Equation (i) represents a parabola with vertex (0,0) and axis as x – axis. Equation (ii) represents a parabola with vertex (0,0) and axis as y - axis.
Points of intersection of the parabola are (0,0) and (4,4).
A rough sketch is given as: -
Now the bounded area is the required area to be calculated, Hence,
Bounded Area, A = [Area between the curve (i) and x axis from 0 to 4] – [Area between the curve (ii) and x axis from 0 to 4]
On integrating the above definite integration,
Area of the region bounded by the parabolasy2 = 4x and x2 = 4y is sq. units.