Parameterize the surface (call it [tex]\mathcal S[/tex]) by
[tex]\mathbf s(u,v)=x(u,v)\,\mathbf i+y(u,v)\,\mathbf j+z(u,v)\,\mathbf k[/tex]
where
[tex]x(u,v)=u[/tex]
[tex]y(u,v)=4u+v^2[/tex]
[tex]z(u,v)=v[/tex]
with [tex]0\le u\le 1[/tex] and [tex]0\le v\le1[/tex]. Then the surface element is
[tex]\mathrm dS=\left\|\mathbf s_u\times\mathbf s_v\|\,\mathrm du\,\mathrm dv=\sqrt{17+4v^2}\,\mathrm du\,\mathrm dv[/tex]
The area of the surface is
[tex]\displaystyle\iint_{\mathcal S}\mathrm dS=\int_{v=0}^{v=1}\int_{u=0}^{u=1}\sqrt{17+4v^2}\,\mathrm du\,\mathrm dv=\int_0^1\sqrt{17+4v^2}\,\mathrm dv[/tex]
[tex]=\dfrac{\sqrt{21}}2+\dfrac{17}4\sinh^{-1}\dfrac2{\sqrt{17}}\approx4.2795[/tex]
