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

The curves y²=4x and x²=4y intersect at the point (4,4).
Area of the square = 4 × 4 = 16
Area between the second curve and the x-axis is ∫₄⁰ (x²/4)dx = (x³/12)₄⁰ = 16/3
Area between the first curve and the y-axis is ∫₄⁰ (y²/4)dy = (y³/12)₄⁰ = 16/3
Therefore, area between the curves must be 16 − (16/3 + 16/3) = 16/3.
Hence, the curves divide the square region into three equal parts.