Answer
$\dfrac{3 \pi}{2}$
Work Step by Step
$div F=\dfrac{\partial (z \tan^{-1} y^2)}{\partial x}+\dfrac{\partial (z^3(\ln (x^2+1))}{\partial y}+\dfrac{\partial (z)}{\partial z}=1$
$\iiint_Ediv \overrightarrow{F}dV=\int_{0}^{2 \pi}\int_0^{1} \int_{1}^{2-r^2} r dz dr d\theta$
or, $=2 \pi \times \int_{0}^{1} [zr] dr$
or, $=2 \pi \times \int_0^{1} r-r^3 dr$
or, $=\dfrac{\pi}{2}$
The flux through the disk is given by $=-\iint_{D} z dA=-\pi(1)^2=-\pi$
Hence the flux through the paraboloid is equal to the total Flux - Flux Through Disk
Thus, we have $Flux=\dfrac{\pi}{2}-(-\pi)=\dfrac{3 \pi}{2}$
