Question

In what ratio does the x-axis divide the area of the region bounded by the parabolas y = 4x – x² and y = x² – x?

Answer 1

Let us find the intersection points first, We have the equations of curves,

y = 4x – x² …….(1)

y = x² – x ………(2)

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

x² – x = 4x – x²

2x² – 5x = 0

x(2x – 5) = 0

x = 0 or x = 5/2

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

At x = 0,

Y = 0² – 0 = 0

At x = 5/2

Hence intersection points are (0, 0) and \(\left(\frac{5}{2},\frac{15}{4} \right)\)

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

Area of Shaded Region = \(\frac{125}{4}\) Square units.