Answer
$\dfrac{3 \pi}{2}$
Work Step by Step
Divergence Theorem: $\iiint_Ediv \overrightarrow{F}dV=\iint_S \overrightarrow{F}\cdot d\overrightarrow{S} $
where, $div F=\dfrac{\partial P}{\partial x}+\dfrac{\partial Q}{\partial y}+\dfrac{\partial R}{\partial z}=\dfrac{\partial (z \tan^{-1} y^2)}{\partial x}+\dfrac{\partial (z^3(\ln (x^2+1))}{\partial y}+\dfrac{\partial (z)}{\partial z}=1$
Divergence Theorem: $\iiint_Ediv \overrightarrow{F}dV=\iint_S \overrightarrow{F}\cdot d\overrightarrow{S} $
$=(2 \pi) \times \int_{0}^{1} [zr] dr$
$=(2 \pi) \times \int_0^{1} (r-r^3) dr$
or, $=\dfrac{\pi}{2}$
The flux through the disk is given by: $-\iint_{D} z dA=-\pi(1)^2=-\pi$
Now, the flux through paraboloid is equal to the (total Flux -Flux Through Disk) , that is, $=\dfrac{\pi}{2}-(-\pi)=\dfrac{3 \pi}{2}$
