Question

Find the area of the surface. the part of the hyperbolic paraboloid z = y2 â x2 that lies between the cylinders x2 + y2 = 4 and x2 + y2 = 25.

Answer 1

$x=u\cos v,y=u\sin v\implies z=u^2\sin^2v-u^2\cos^2v=-u\cos2v$

$\mathbf s(u,v)=\langle u\cos v,u\sin v,-u\cos2v\rangle$

$\mathrm dS=\|\mathbf s_u\times\mathbf s_v\|=u\sqrt{1+4u^2}\,\mathrm du\,\mathrm dv$

$\displaystyle\iint_{\mathcal S}\mathrm dS=\int_{v=0}^{v=2\pi}\int_{u=2}^{u=5}u\sqrt{1+4u^2}\,\mathrm du\,\mathrm dv=\dfrac{(101^{3/2}-17^{3/2})\pi}6$