Question

Find the area of the surface. The part of the paraboloid z = x2 + y2 that lies inside the cylinder x2 + y2 = 9.

Answer 1

Parameterize the surface $S$ by

$\mathbf s(r,\theta)=(x(r,\theta),y(r,\theta),z(r,\theta))=(r\cos\theta,r\sin\theta,r^2)$

with $0\le r\le3$ and $0\le\theta\le2\pi$.

The area of $S$ is then given by the surface integral

$\displaystyle\iint_S\mathrm dS=\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=3}\left\|\mathbf s_r\times\mathbf s_\theta\right\|\,\mathrm dr\,\mathrm d\theta$
$=\displaystyle\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=3}r\sqrt{1+4r^2}\,\mathrm dr\,\mathrm d\theta$
$=\displaystyle\dfrac\pi6(1+4r^2)^{3/2}\bigg|_{r=0}^{r=3}$
$=\dfrac{(37\sqrt{37}-1)\pi}6$