Answer
$\dfrac{\pi a^3}{6}$
Work Step by Step
We have: $|\nabla g|=\sqrt {4x^2+4y^2 +4z^2 }= 2a$
and $\iint_{S} F \cdot n=\dfrac{z^2}{a}$
We set up the integral and solve the flux of $F$ as follows:
$\iint_{S} F \cdot n \ d \theta =\iint_{R} (\dfrac{z^2}{a}) (\dfrac{a}{z}) \ dA$
or, $=\iint_{R} \ z \ dA$
or, $=\int_{0}^{\pi/2} \int_{0}^{a} \sqrt {a^2-(x^2+y^2)} \ dx \ dy$
or, $= \int_{0}^{\pi/2} \int_0^a \sqrt {a^2-r^2} \ dr \ d \theta $
or, $= \dfrac{\pi a^3}{6}$
