Find the area of the region bounded by the curve y = √1 – x^{2}, line y = x and the positive x - axis.

To find the area of the region bounded by

y = √1 – x^{2} …(i)

X^{2} + y^{2} = 1

X = y …(ii)

On solving the equation (i) and (ii),

Or X^{2} + x^{2} = 1

Or 2X^{2} = 1

Or x =

Equation (i) represents a circle (0,0) and meets axes at (±1,0), (0,±1).

Equation (ii) represents a line passing through B and and they are also points of intersection.

These are shown in the graph below:

Required area = Region OABO

= Region OCBO + Region CABC

The area of the region bounded by the curve y = √1 – x^{2}, line y = x is

40