Prove that the curves y² = 4x and x² = 4y divide the area of the square bounded by x = 0, x = 4, y = 4 and y = 0 into three equal parts.

The equation of curves are
                                                 ...(1)
                  and                        ...(2)
From (2),                                   ...(3)
Putting this value of y in (1),

or   


From P. draw PM ⊥ x-axis.
Required area = Area OAPB = Area OBPM - area OAPM
                        
Now, the area of the region OAQBO bounded by curves  and 

                                 
Again, the area of the region OPQAO bounded by the curves x² = 4 y, x = 0, x = 4 and x-axis

Similarly, the area of the region OBQRO bounded by the curve 
    ...(3)
From (1), (2) and (3), it is clear that the area of the region OAQBO = area of the region OPQAO = area of the region OBQRO, i.e., urea bounded by parabolas y² = 4 x and x² = 4 y divides the area of the square in three equal parts.