The plane [tex]z=2-x-y[/tex] has vertices at the intercepts (2, 0, 0), (0, 2, 0), and (0, 0, 2). Parameterize the tetrahedral face (call it [tex]T[/tex]) by
[tex]\vec s(u,v)=(1-v)((1-u)\langle2,0,0\rangle+u\langle0,2,0\rangle)+v\langle0,0,2\rangle[/tex]
[tex]\vec s(u,v)=\langle2(1-u)(1-v),2u(1-v),2v\rangle[/tex]
for [tex]0\le u\le1[/tex] and [tex]0\le v\le1[/tex]. Take the normal vector to [tex]T[/tex] to be
[tex]\dfrac{\partial\vec s}{\partial u}\times\dfrac{\partial\vec s}{\partial v}=4(1-v)\langle1,1,1\rangle[/tex]
Then the flux of [tex]\vec F[/tex] across [tex]T[/tex] is
[tex]\displaystyle\iint_T\vec F\cdot\mathrm d\vec S=4\int_0^1\int_0^1(1-v)\langle0,0,-2\rangle\cdot\langle1,1,1\rangle\,\mathrm du\,\mathrm dv[/tex]
[tex]=\displaystyle-8\int_0^1(1-v)\,\mathrm dv=\boxed{-4}[/tex]
