I know that there's a typo and x and p refer to the same thing. But what I don't understand is what exactly x represents. It must be the point on the surface of the object at which our ray hits right? The weird thing is that a few slides from here the pseudocode has a line that says intersect(x, wi). But it makes no sense to intersect a line with a point. So does x just refer to the entire object?

Ok nevermind, I see that x just refers to a point position. The psuedocode in a few slides from here that says intersect(x, wi) is just odd notation for "find me the point on the light source that intersects with the ray $\omega_i$ that extends from $x$. This was we get the radiance from the light source along $\omega_i$.

I didn't really understand this graph until I figured out what this polar coordinates system meant.

Fundamentally, what we are doing is for a point calculating the irradiance values from light sources from all different angles within a hemispherical zone.

To iterate over the integral, we use polar coordinates omega phi, to find each incidental ray within the hemisphere, which is equivalent to the colloquial description above.