Question

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

Answer 1

The required area is bounded between two curves y = x² and y = |x|. Both of these curves are symmetric about y-axis and shaded region in the fig. shows the region whose area is required.

Therefore, the required area

A = 2 x Area of the region R₁ 

Now, to find the point of intersection of the curves y = |x| and y = x² , we solve them simultaneously.

Clearly, the region R₁ is in the first quadrant, where x > 0,

Solving these two equations, we get

The limits are, when x = 0, y = 0 and when x = 1, y = 1. So, the point of intersection of the curves are O(0, 0) and A(1, 1). 

Now, required Area = 2 x Area of line region R₁