In what ratio does the x-axis divide the area of the region bounded by the parabolas y = 4x – x^{2} and y = x^{2} – x?

Let us find the intersection points first,

We have the equations of curves,

y = 4 x – x^{2} …….(1)

y = x^{2} – x ………(2)

From (1) and (2) we can get,

x^{2} – x = 4 x – x^{2}

2 x^{2} – 5x = 0

x(2 x – 5) = 0

x = 0 or x = 5/2

Putting these values of x in equation (2) we get,

At x = 0,

Y = 0^{2} – 0 = 0

At x = 5/2

Hence intersection points are (0, 0) and

Area bounded by the curves is shown by the shaded region of the figure shown above.

Area of shaded region =

Area of shaded region =

Area of Shaded Region =

Area of Shaded Region = Square units.

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