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


...(i)


And x2 = 4y


...(ii)


On solving the equation (i) and (ii),


= 4x


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.


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