Parameterize the surface [tex]\mathcal S[/tex] by
[tex]\mathbf s(u,v)=(\langle1,0,0\rangle(1-u)+\langle0,6,0\rangle u)(1-v)+\langle0,0,6\rangle v[/tex]
[tex]\mathbf s(u,v)=\langle(1-u)(1-v),6u(1-v),6v\rangle[/tex]
with [tex]0\le u\le1[/tex] and [tex]0\le v\le1[/tex], which has surface element
[tex]\mathrm dS=\|\mathbf s_u\times\mathbf s_v\|\,\mathrm du\,\mathrm dv=6\sqrt{38}(1-v)\,\mathrm du\,\mathrm dv[/tex]
Then the surface integral becomes
[tex]\displaystyle\iint_{\mathcal S}xy\,\mathrm dS=\int_{u=0}^{u=1}\int_{v=0}^{v=1}(1-u)(1-v)(6u(1-v))(6\sqrt{38}(1-v))\,\mathrm dv\,\mathrm du=3\sqrt{\dfrac{19}2}[/tex]
