$$ $$

$ S = 4 \int_0^{\frac{\pi}{4} } \sqrt { r^2 + (\frac{dr}{d\theta})^2 } d\theta \\ $

$ = 4 \int_0^{\frac{\pi}{4} } \sqrt { a^2cos2\theta + (\frac{dr}{d\theta})^2 } d\theta \\ $

$ r = a\sqrt{cos2\theta } \\ $

$ \frac{dr}{d\theta} = a \frac{1}{2\sqrt{cos2\theta}} * -2sin2\theta \\ $

$ r^2 + (\frac{dr}{d\theta})^2 = a^2 cos2\theta + \frac{a^2sin^22\theta}{cos2\theta } \\ $

$ = \frac{a^2}{cos2\theta} \\ $

$ S = 4 \int_0^{\frac{\pi}{4}} \sqrt{\frac{a^2}{cos2\theta}} d\theta \\ $

$ = 4a \int_0^{\frac{\pi}{4}} \sqrt{\frac{1}{cos2\theta}} d\theta \\ $

$ 2\theta = t \Longrightarrow 2d\theta = dt \\ $

$ S = \frac{4a}{2} \int_0^{\frac{\pi}{2}} \sqrt{\frac{dt}{cost}} \\ $

$ = \int_0^{\frac{\pi}{4}} sin^0t cos^{\frac{-1}{2}t} dt \\ $

$ = a \beta(\frac{1}{2}, \frac{\frac{-1}{2} + 1}{2} ) = a \beta(\frac{1}{2}, \frac{1}{4} ) \\ $

$ = a\frac{ \Gamma \frac{1}{2} \Gamma \frac{1}{2} } {\Gamma(\frac{1}{2} + \frac{1}{4}) } \\ $

$ = \frac { a\sqrt{\pi} \Gamma \frac{1}{4} * \Gamma \frac{1}{4}} {\Gamma \frac{3}{4} * \Gamma \frac{1}{4} } \\ $

$ = \frac { a\sqrt{\pi} (\Gamma \frac{1}{4} )^2} {\frac{\pi}{sin\frac{\pi}{4}}} \\ $

$ = \frac { a\sqrt{\pi} (\Gamma \frac{1}{4} )^2} {\pi \sqrt2} \\ $

$ = \frac { a (\Gamma \frac{1}{4} )^2} {\sqrt{\pi} \sqrt2} \\ $

$ = \frac { a (\Gamma \frac{1}{4} )^2} {\sqrt{2\pi}} \\ $