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*\mathbf s_v\|\,\mathrm du\,\mathrm dv=u√(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√(1+4u^2)\,\mathrm du\,\mathrm dv=\frac{\left((1+4r^2)^(3/2)-1\right)\pi}6`