Looks like
[tex]\vec F(x,y,z)=y\,\vec\imath+z\,\vec\jmath+xz\,\vec k[/tex]
The solid described by [tex]x^2+y^2\le z\le 1[/tex] is the interior of the region bounded by the paraboloid [tex]z=x^2+y^2[/tex] and the plane [tex]z=1[/tex]. Call this region [tex]R[/tex].
By the divergence theorem,
[tex]\displaystyle\iint_{\partial R}\vec F\cdot\mathrm d\vec S=\iiint_R\mathrm{div}\vec F\,\mathrm dV[/tex]
The divergence is
[tex]\mathrm{div}\vec F(x,y,z)=(y)_x+(z)_y+(xz)_z=x[/tex]
so that the surface integral reduces to
[tex]\displaystyle\iiint_Rx\,\mathrm dV[/tex]
or in cylindrical coordinates,
[tex]\displaystyle\int_0^{2\pi}\int_0^1\int_{r^2}^1r^2\cos\theta\,\mathrm dz\,\mathrm dr\,\mathrm d\theta=\boxed{0}[/tex]
