I have a difficulty to understand when I should use the surface integral over a surface or the triple integral over a space. For instance:
If $S$ is the subset of the sphere with radius $1$ with $x>0$, then $\int_{S} xz\ dx\ dy\ dz = $?
I tried both surface and triple integral and I found the same answer, but which integral is the correct? We have sphere, which means $x^{2} + y^{2} + z^{2} = 1$ , not ball $(x^{2} + y^{2} + z^{2} \leq 1)$ so I guess that surface integral is the correct one. Am I right? Any opinions?