Answer
$$\dfrac{\pi a^3}{6}$$
Work Step by Step
Since, $|\nabla g|=\sqrt {4x^2+4y^2 +4z^2 }= 2a$
and $\iint_{S} F \cdot n=\dfrac{z^2}{a}$
Solve the flux of $F$.
$$\iint_{S} F \cdot n \ d \theta =\iint_{R} (\dfrac{z^2}{a}) \times (\dfrac{a}{z}) \ dA \\=\iint_{R} ( z) dA \\=\int_{0}^{\pi/2} \int_{0}^{a} \sqrt {a^2-(x^2+y^2)} \ dx \ dy \\= \int_{0}^{\pi/2} \int_0^a \sqrt {a^2-r^2} \ dr \ d \theta \\= \dfrac{\pi a^3}{6}$$
