Find the area of the region common to the parabolas 4y^{2} = 9x and 3 x^{2} =16y.

The given equations are,

4y^{2} = 9x

And 3x^{2} = 16y.

Equating equation (i) and (ii)

Or

Or x = 4

When we put x = 4 in equation (i) then y = 3,

When we put x = 0 in equation (i) then y = 0,

On solving these two equations, we get the point of intersections.

The points are O (0, 0) and A(4,3). These are shown in the graph below

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,

The required area =

Area of the region common to the parabolas 4y^{2} = 9x and 3 x^{2 =} 16y is 4 sq. units

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