You can use the divergence theorem:
[tex]\vec v=z\,\vec\imath+y^2\,\vec\jmath+x^2\,\vec k[/tex]
has divergence
[tex]\mathrm{div}\vec v=\dfrac{\partial z}{\partial x}+\dfrac{\partial y^2}{\partial y}+\dfrac{\partial x^2}{\partial z}=2y[/tex]
Then the rate of flow out of the cylinder (call it R) is
[tex]\displaystyle\iint_{\partial R}\vec v\cdot\mathrm d\vec S=\iiint_R\mathrm{div}\vec v\,\mathrm dV[/tex]
(by divergence theorem)
[tex]=\displaystyle2\int_0^{2\pi}\int_0^5\int_0^1r^2\sin\theta\,\mathrm dz\,\mathrm dr\,\mathrm d\theta[/tex]
(after converting to cylindrical coordinates)
whose value is 0.
