Let $\mathbf{v}=\left< 3 x,0,6 z \right>$ be the velocity field (in $\mathrm{ft/s}$) of a fluid in $\mathbf{R}^3$.
Calculate the flow rate (in $\mathrm{ft^3/s}$) through the upper hemisphere of the sphere $x^2+y^2+z^2=1$ $(z\ge 0)$.

$\iint_S \mathbf{v} \cdot d\mathbf{S}=$ ${\mathrm{ft}}^3/\mathrm{s}$