Find the area enclosed by the parabolas y = 4x – x^{2} and y = x^{2} – x.

To area enclosed by

Y = 4x – x^{2} (1)

4x – x^{2} = x^{2} – x

2x^{2} – 5x = 0

x = 0 or x =

y = 0 or y =

And y = x^{2} – x

…(2)

Equation (1) represents a parabola downward with vertex at (2,4) and meets axes at (4,0), (0,0).

Equation (2) represents a parabola upward whose vertex is and meets axes at Q(1,0), (0,0).Points of intersection of parabolas are O (0,0) and A .

These are shown in the graph below:

Required area = Region OQAP

The area enclosed by the parabolas y = 4x – x^{2} and y = x^{2} – x is .

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