1. ## surface integral help!

Evaluate the surface integral the integral of S of (x+y+z)ds, where S is the hemisphere x^2+y^2+z^2 = a^2 at z> or = 0.

Thank you very much.

2. Originally Posted by kittycat
Evaluate the surface integral the integral of S of (x+y+z)ds, where S is the hemisphere x^2+y^2+z^2 = a^2 at z> or = 0.

Thank you very much.
We can parametrize the surface, $(r,\theta)\mapsto (r\cos \theta, r\sin\theta, \sqrt{a^2-r^2})$ where $0\leq \theta \leq 2\pi$ and $0\leq r\leq a$, let us call that parametrization by $\bold{G}(r,\theta)$ and the function $(x,y,z)\mapsto x+y+z$ as $f(x,y,z)$. That means the surface integral is,
$\iint_R f\left( \bold{G}(r,\theta) \right) \cdot \left| \frac{\partial \bold{G}}{\partial r}\times \frac{\partial \bold{G}}{\partial \theta} \right| dA$ where $R$ is the rectangle $0\leq \theta \leq 2\pi$ and $0\leq r\leq a$. Can you finish that?