Parameterize [tex]T[/tex] by
[tex]\vec s(u,v)=(1-v)\bigg((1-u)(7\,\vec\imath)+u(7\,\vec\jmath\bigg)+v(7\,\vec k)=7(1-u)(1-v)\,\vec\imath+7u(1-v)\,\vec\jmath+7v\,\vec k[/tex]
with both [tex]u,v[/tex] ranging from 0 to 1.
The fluid flows with velocity [tex]\vec v=2\,\vec k[/tex], so it's flowing upward and we should take the normal vector to [tex]T[/tex] to be pointing away from the origin:
[tex]\dfrac{\partial\vec s}{\partial u}\times\dfrac{\partial\vec s}{\partial v}=(49-49v)(\vec\imath+\vec\jmath+\vec k)[/tex]
Then the flow rate of [tex]\vec v[/tex] across [tex]T[/tex] is
[tex]\displaystyle\iint_T\vec v\cdot\mathrm d\vec S=\int_0^1\int_0^1(49-49v)(2\,\vec k)\cdot(\vec\imath+\vec\jmath+\vec k)\,\mathrm du\,\mathrm dv[/tex]
[tex]=\displaystyle98\int_0^1\int_0^1(1-v)\mathrm du\,\mathrm dv=\boxed{49}[/tex]
(in m/s)
The projection of [tex]T[/tex] on the [tex]x,y[/tex] plane should have vertices (7, 0, 0), (0, 7, 0), and (0, 0, 0). (unless you're talking about another [tex]T[/tex] altogether). Parameterize [tex]T[/tex] by
[tex]\vec s(u,v)=(1-v)\bigg((1-u)(7\vec\imath)+u(7\,\vec\jmath)\bigg)=7(1-u)(1-v)\,\vec\imath+7u(1-v)\,\vec\jmath[/tex]
again with both [tex]u,v[/tex] between 0 and 1.
The normal vector is then
[tex]\dfrac{\partial\vec s}{\partial u}\times\dfrac{\partial\vec s}{\partial v}=(49-49v)\,\vec k[/tex]
and so the flow rate is
[tex]\displaystyle\iint_T\vec v\cdot\mathrm d\vec S=\int_0^1\int_0^1(49-49v)(2\,\vec k)\cdot(\vec k)\,\mathrm du\,\mathrm dv[/tex]
[tex]=\displaystyle98\int_0^1\int_0^1(1-v)\,\mathrm du\,\mathrm dv=\boxed{49}[/tex]
