Question

Find the area of the region {(x, y): x² ≤ y ≤ x}

Answer 1

We have, y = x² ....(i)

and y = x ....(ii)

We know that y = x² is an upward parabola and the line y = x is passing through origin.

Now, on solving (i) and (ii), we get

x² = x ⇒ x(x - 1)

x = 0 or 1

From(ii), x = 0 ⇒ y = 0 and x = 1 ⇒ y = 1

So, the point of intersection of (i) and (ii) are O(0, 0) and A (1, 1).

Draw AB ⊥ OX

Required area = Shaded area shown in figure

= area OPABO - area OQABO

Hence, the required area is \(\frac 16\) sq unit.