Question

Find the area of the surface. the part of the hyperbolic paraboloid z = y2 − x2 that lies between the cylinders x2 + y2 = 16 and x2 + y2 = 25.

Answer 1

Parameterize the surface (call it $\mathcal S$) by

$\mathbf s(u,v)=\langle u\cos v,u\sin v,u^2\sin^2v-u^2\cos^2v\rangle=\langle u\cos v,u\sin v,-u^2\cos2v\rangle$

with $4\le u\le5$ and $0\le v\le2\pi$, which yields

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

Then the area of $\mathcal S$ is given by the surface integral

$\displaystyle\iint_{\mathcal S}\mathrm dS=\int_{v=0}^{v=2\pi}\int_{u=4}^{u=5}u\sqrt{1+4u^2}\,\mathrm du\,\mathrm dv=\dfrac{(101^{3/2}-65^{3/2})\pi}6$