Same parameterization as in the provided link, which I'll reproduce correctly here:
[tex]\mathbf s(u,v)=\langle u,3\cos v,3\sin v\rangle[/tex]
and this time [tex]0\le u\le5[/tex] and [tex]\cos^{-1}\dfrac23\le v\le\dfrac\pi2[/tex]
We get the same surface element,
[tex]\mathrm d\mathbf S=3\,\mathrm du\,\mathrm dv[/tex]
and the area is
[tex]\displaystyle3\iint_{v=\cos^{-1}(2/3)}^{v=\pi/2}\int_{u=0}^{u=5}\mathrm du\,\mathrm dv=\dfrac{15\pi}2-15\cos^{-1}\dfrac23[/tex]
