Answer
$$\dfrac{\pi}{3}$$
Work Step by Step
Apply cylindrical coordinates. $ x=r \cos \theta ;\\ y= r \sin \theta ;\\ z \gt 0$
We know that $ r(r, \theta)=xi+yj+zk $ or, $ r^2=x^2+y^2+z^2$
$$ Surface \space Area =\iint_{R} \dfrac{|\nabla f|}{|\nabla f \cdot p|} \space dA\\=(2) \times \space \int_{-1/2}^{1/2} \int_{0}^{1/2}\dfrac{1}{\sqrt {1-x^2} } \space dy \space dx \\=[\sin^{-1}x]_{-1/2}^{1/2} \\=\dfrac{\pi}{3}$$
