With no specific integrand given, I'm assuming you need only find the area of [tex]\mathcal S[/tex], in which case we can parameterize the surface by
[tex]\mathbf s(u,v)=(\langle1,0,0\rangle(1-u)+\langle0,5,0\rangle u)(1-v)+\langle0,0,5\rangle v[/tex]
[tex]\mathbf s(u,v)=\langle(1-u)(1-v),5u(1-v),5v\rangle[/tex]
with [tex]0\le u\le1,0\le v\le1[/tex]. Then the area of [tex]\mathcal S[/tex] is given by
[tex]\displaystyle\iint_{\mathcal S}\mathrm dS=\int_{[0,1]^2}\|\mathbf s_u\times\mathbf s_v\|\,\mathrm dv\,\mathrm du[/tex]
[tex]\displaystyle=15\sqrt3\int_{u=0}^{u=1}\int_{v=0}^{v=1}(1-v)\,\mathrm dv\,\mathrm du=\frac{15\sqrt3}2[/tex]
