Let $T$ be the figure bounded by: $y-\sqrt{1-x^2-z^2}=0$, $x^2+z^2=1$, $y-z=2$(green axis is y, blue - z)
I need calculate the lateral flow for semi-sphere(that burgundy thing on the image) $\vec a = ..\vec i+..\vec j+..\vec k$(I will not write the formula since it is not important in the question).
What I've done?
First of all, I tried to count general flux. I use that $\int\int(\vec a, \vec n)dT=\int\int\int( div\vec a )dx dy dz$. $div\vec a=const$. The main problem is that I can't even imagine, how to set limits of integration in this triple integral. I tried cylindrical coordinate system, where $x=rcos(\phi), y=y, z=rsin(\phi)$, but the same trouble.
The cylindrical coordinates that you chose later is the right approach. If you see the region, $y$ is bound between the sphere and the plane. $x, z$ is bound by cylinder $x^2+z^2 = 1$.
So limits of integration is,
$\sqrt{1-x^2-z^2} \leq y \leq {z+2}, \ 0 \leq x^2+z^2 \leq 1$.
So using cylindrical coordinates as,
$x = r \cos\theta, y = y, z = r\sin\theta$
The limits of integration is,
$\sqrt{1-r^2} \leq y \leq r \sin\theta + 2, 0 \leq r \leq 1, 0 \leq \theta \leq 2\pi$