I'm trying to evaluate the flux through hemisphere and reached this: $$ \int_0^{0.5\pi} \int_0^{2\pi} \frac{I \cos(\phi)}{r^2} r^2 \sin(\theta) ~ d\phi d\theta \\ $$
The limits chosen above give the right values I needed after integration.
My question is, why can't I use the limits 0<ϕ<π and 0<θ<π ? Since that seems to give a zero when I try to integrate it, but when I try to visualize it as area swept it seems to give a hemisphere.
EDIT: I'm using the convention where θ is angle from azimuth