Let $S$ be the half-hemisphere $ x^2 +y^2 + z^2 = 1, y \ge 0, z \ge 0 $
The boundaries for the above question are $$ 0 \lt \theta \lt \frac \pi 2 , 0 \lt \varphi \lt \pi$$ This example tells me to find the flux across S. I can set up the integration but I can't work out how to obtain the boundaries. I've tried drawing it up but I can't see it.
Another example is
Let A be the surface $x^2 +y^2 +z^2 = 100, z \gt 0$
The boundaries stated are $$ 0 \lt \theta \lt \frac \pi 2 , 0 \lt \varphi \lt 2\pi$$
Thank you in advance!
You are working with spherical coordinates using the typical Physics notation:$$x=r\cos\varphi\sin\theta\text{, }y=r\sin\varphi\sin\theta\text{, and }z=r\cos\theta,$$with $\varphi\in[0,2\pi]$, and $\theta\in[0,\pi]$.
In your first example, you want to have $y,z\geqslant0$. This means that $\sin\varphi\sin\theta,\cos\theta\geqslant0$. Since $\theta\in[0,\pi]$, $\sin\theta\geqslant0$, but $\cos\theta\geqslant0\iff\theta\in\left[0,\frac\pi2\right]$. On the other hand, $\sin\varphi\geqslant0\iff\varphi\in[0,\pi]$.
Can you work out the other example now?