I see that we are integrating over the hemisphere, but what exactly does that mean?
I think the integration over the Hemisphere is done so we can calculate the total Irradiance from the Environment on a surface. Any light coming from the other hemisphere would be blocked by the surface. I'm not 100% sure though
When we do calculations on the irradiance of the lights, the intensity of some lights might be too low to make a significant impact on the result of a certain point. If we really try to include all points, it might be a waste of time since the effect of some of them are subtle.