Find the surface integral $\int\limits_\sigma\!\!\int f(x,y,z)\,dS$ where $f(x,y,z) = \left(x^{2}+y^{2}\right)z$ and $\sigma$ is the sphere $x^2+y^2+z^2 = 9$ above $z = 2$.
Parameterize the surface integral

$\displaystyle\int\limits_\sigma\!\!\int f(x,y,z)\,dS = \int\limits_R\!\!\int f\left(x(\theta,\phi),y(\theta,\phi),z(\theta,\phi)\right)\left\Vert\frac{\partial r}{\partial \theta}\times\frac{\partial r}{\partial \phi}\right\Vert dA$

=

$\displaystyle\int$

$\displaystyle\int$

$\Bigg\Vert$

$\Bigg\Vert\hskip 3pt d\theta\, d\phi$

Note: For $\theta$ type theta and for $\phi$ type phi.