[tex]x=u\cos v,y=u\sin v\implies z=u^2\sin^2v-u^2\cos^2v=-u\cos2v[/tex]
[tex]\mathbf s(u,v)=\langle u\cos v,u\sin v,-u\cos2v\rangle[/tex]
[tex]\mathrm dS=\|\mathbf s_u\times\mathbf s_v\|=u\sqrt{1+4u^2}\,\mathrm du\,\mathrm dv[/tex]
[tex]\displaystyle\iint_{\mathcal S}\mathrm dS=\int_{v=0}^{v=2\pi}\int_{u=2}^{u=5}u\sqrt{1+4u^2}\,\mathrm du\,\mathrm dv=\dfrac{(101^{3/2}-17^{3/2})\pi}6[/tex]
