Parameterize the part of the surface we care about, denoted [tex]\mathcal S[/tex], by
[tex]\mathbf r(u,v)=\langle u,5u+v^2,v\rangle[/tex]
with [tex]0\le u\le1[/tex] and [tex]0\le v\le1[/tex].
Then
[tex]\|\mathbf r_u\times\mathbf r_v\|=\|\langle5,-1,2v\rangle\|=\sqrt{4v^2+26}[/tex]
The area is given by the surface integral
[tex]\displaystyle\iint_{\mathcal S}\mathrm dS=\int_{v=0}^{v=1}\int_{u=0}^{u=1}\|\mathbf r_u\times\mathbf r_v\|\,\mathrm du\,\mathrm dv[/tex]
[tex]=\displaystyle\int_{v=0}^{v=1}\sqrt{4v^2+26}\,\mathrm dv=\sqrt{\dfrac{15}2}+\dfrac{13}2\sinh^{-1}\sqrt{\dfrac2{13}}[/tex]
