The given equations are,

And y = x ...(ii)

Solving equation (i) and (ii)

= x = y

Or = y

Or y(y – 1) = 0

So, y = 0 or y = 1 and x = 0 or x = 1

On solving these two equations, we get the points of intersection.

The points are O (0, 0) and A(1,1). These are shown in the graph below

Now the bounded area is the required area to be calculated,

Hence, Bounded Area, A = [Area between the curve (i) and x axis from 0 to 1] - [Area between the curve (ii) and x axis from 0 to 1]

On integrating the above definite integration,

Area of the region bounded by and Y = X is.