Answer
$$A = 2\sqrt 3 - \frac{{2\pi }}{3}$$
Work Step by Step
$$\eqalign{
& y = \frac{1}{{\sqrt {1 - {x^2}} }},{\text{ and }}y = 2 \cr
& {\text{Let }}y = y \cr
& 2 = \frac{1}{{\sqrt {1 - {x^2}} }} \cr
& 2\sqrt {1 - {x^2}} = 1 \cr
& 4\left( {1 - {x^2}} \right) = 1 \cr
& 4 - 4{x^2} = 1 \cr
& 4{x^2} = 3 \cr
& x = \pm \sqrt {\frac{3}{4}} \cr
& x = \pm \frac{{\sqrt 3 }}{2} \cr
& {\text{From the graph shown below we have that:}} \cr
& A = \int_{ - \frac{{\sqrt 3 }}{2}}^{\frac{{\sqrt 3 }}{2}} {\left( {2 - \frac{1}{{\sqrt {1 - {x^2}} }}} \right)} dx \cr
& {\text{By symmetry}} \cr
& A = 2\int_0^{\frac{{\sqrt 3 }}{2}} {\left( {2 - \frac{1}{{\sqrt {1 - {x^2}} }}} \right)} dx \cr
& {\text{Integrating}} \cr
& A = 2\left[ {2x - {{\sin }^{ - 1}}x} \right]_0^{\frac{2}{{\sqrt 3 }}} \cr
& A = 2\left[ {2\left( {\frac{{\sqrt 3 }}{2}} \right) - {{\sin }^{ - 1}}\frac{{\sqrt 3 }}{2}} \right] - 2\left[ {2\left( 0 \right) - {{\sin }^{ - 1}}0} \right] \cr
& {\text{Simplifying}} \cr
& A = 2\left[ {\sqrt 3 - \frac{\pi }{3}} \right] - 2\left[ 0 \right] \cr
& A = 2\sqrt 3 - \frac{{2\pi }}{3} \cr} $$
