[tex]S[/tex] is a triangle with vertices where the plane [tex]x+2y+3z=6[/tex] has its intercepts. These occur at the points (0,0,2), (0,3,0), and (6,0,0). Parameterize [tex]S[/tex] by
[tex]\vec s(u,v)=(1-v)((1-u)(0,0,2)+u(0,3,0))+v(6,0,0)[/tex]
[tex]\vec s(u,v)=(6v,3u(1-v),2(1-u)(1-v))[/tex]
with [tex]0\le u\le1[/tex] and [tex]0\le v\le1[/tex]. The surface element is
[tex]\mathrm d\sigma=\left\|\dfrac{\partial\vec s}{\partial u}\times\dfrac{\partial\vec s}{\partial u}\right\|\,\mathrm du\,\mathrm dv=6\sqrt{14}(1-v)\,\mathrm du\,\mathrm dv[/tex]
So the integral is
[tex]\displaystyle\iint_Sxyz\,\mathrm d\sigma=216\sqrt{14}\int_0^1\int_0^1uv(1-u)(1-v)^3\,\mathrm du\,\mathrm dv[/tex]
