By the divergence theorem, the integral of [tex]\vec F[/tex] over the surface [tex]S[/tex] is equivalent to the integral of the divergence of [tex]\vec F[/tex] over the interior of [tex]S[/tex] (call it [tex]R[/tex]).
The divergence of [tex]\vec F[/tex] is
[tex]\nabla\cdot\vec F=6xyz[/tex]
so the surface integral is equivalent to
[tex]\displaystyle\iint_S\vec F\cdot\mathrm d\vec S=\iiint_R(\nabla\cdot\vec F)\,\mathrm dV[/tex]
[tex]=\displaystyle6\int_0^c\int_0^b\int_0^axyz\,\mathrm dx\,\mathrm dy\,\mathrm dz=\boxed{\frac34a^2b^2c^2}[/tex]
