This is a problem from Chennai Mathematical Institute, CMI Entrance 2014 based on area of a region. Try to solve it.

Problem: Area of a region

. Find the area of A.

Discussion:

is a disc of radius 12 with center (0, 0).

Now notice that

Hence if a point (x, y) is in A then (-x, -y) is not in A. Similarly if there is a point (x, y) not in A then we get a corresponding point (-x, -y) in A.

Therefore we have a bijection between points in A and not in A.

Thus area of A is exactly half the area of the disc =

This can in fact be generalised to any odd function f(x). f(ax+by) divides the area of the disc in half if f is odd.