= Solution
Use the full field in the PDF, including the exponential term missing from the transcription. Its <divergence> is
$$
\nabla\cdot F=[z\sin(yz)+3x^2]-z\sin(yz)+3y^2=3(x^2+y^2).
$$
The exponential in the third component has zero $z$ <derivative>. In <cylindrical coordinates>, the solid has $0\leq r\leq2$ and $0\leq z\leq4-r^2$. Therefore the <divergence theorem> gives the outward <flux>
$$
\begin{aligned}
\int_S F\cdot\hat n\,dS&=\int_0^{2\pi}\int_0^2\int_0^{4-r^2}3r^2\,r\,dz\,dr\,d\phi\\
&=6\pi\int_0^2(4r^3-r^5)\,dr=\boxed{32\pi}.
\end{aligned}
$$
The <integration> includes the entire closed boundary, both cap and base.
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