Parameterize the surface [tex]S[/tex] by
[tex]\mathbf s(r,\theta)=(x(r,\theta),y(r,\theta),z(r,\theta))=(r\cos\theta,r\sin\theta,r^2)[/tex]
with [tex]0\le r\le3[/tex] and [tex]0\le\theta\le2\pi[/tex].
The area of [tex]S[/tex] is then given by the surface integral
[tex]\displaystyle\iint_S\mathrm dS=\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=3}\left\|\mathbf s_r\times\mathbf s_\theta\right\|\,\mathrm dr\,\mathrm d\theta[/tex]
[tex]=\displaystyle\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=3}r\sqrt{1+4r^2}\,\mathrm dr\,\mathrm d\theta[/tex]
[tex]=\displaystyle\dfrac\pi6(1+4r^2)^{3/2}\bigg|_{r=0}^{r=3}[/tex]
[tex]=\dfrac{(37\sqrt{37}-1)\pi}6[/tex]
