Question

Find the area of the region bounded by the curve y = √1 – x², line y = x and the positive x - axis.

Answer 1

To find the area of the region bounded by

y = √1 – x² …(i)

x² + y² = 1

x = y …(ii) On solving the equation (i) and (ii),

Or x² + x² = 1

Or 2x² = 1

Or x = \(\pm \frac{1}{\sqrt{2}}\)

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

Equation (ii) represents a line passing through B \(\left(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}} \right)\) and \(\left(-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}} \right)\) 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 – x², line y = x is \(\frac{\pi}{8}\) sq. units.