Question

Find the area of the part of the circular paraboloid z = x2 + y2 where 0 x2 + y2 r2

Answer 1

Parameterize the parabolic "cup" (call it $\mathcal S$) by

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

with $0\le u\le r$ and $0\le v\le2\pi$. Then the surface element is

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

and the area is given by the surface integral

$\displaystyle\iint_{\mathcal S}\mathrm dS=\int_{v=0}^{v=2\pi}\int_{u=0}^{u=r}u\sqrt{1+4u^2}\,\mathrm du\,\mathrm dv=\frac{\left((1+4r^2)^{3/2}-1\right)\pi}6$