The area of the region $$R = \left\{ {\left( {x,\,y} \right):\left| x \right| \leqslant \left| y \right|{\text{ and }}{x^2} + {y^2} \leqslant 1} \right\}$$        is :

Solution :
Required area $$=$$ area of the shaded region
$$ = 4$$  (area of the shaded region in first quadrant
$$\eqalign{ & = 4\int_0^{\frac{1}{{\sqrt 2 }}} {\left( {{y_1} - {y_2}} \right)} dx \cr & = 4\int_0^{\frac{1}{{\sqrt 2 }}} {\left( {\sqrt {1 - {x^2}} - x} \right)dx} \cr & = 4\left[ {\frac{1}{2} \times \sqrt {1 - {x^2}} + \frac{1}{2}{{\sin }^{ - 1}}x - \frac{{{x^2}}}{2}} \right]_0^{\frac{1}{{\sqrt 2 }}} \cr & = \frac{\pi }{2}{\text{ sq}}{\text{. unit}} \cr} $$