Rendering: Clearing Doubts About The Rendering Equation

From what I understand, the rendering equation can be considered in two ways:

  1. as a double integral with respect to the horizontal and vertical angles in a hemisphere, in spherical coordinates
  2. as a (single) integral where the differential is dw, a differential solid angle.

The rendering equation is recursive, because the incoming light can be thought of as the outgoing light from a different surface, thus, the rendering equation accounts for indirect illumination as well.

Due to the continuous nature of the equation, in order to solve it in realtime, one has to discretize it by breaking it up into a bunch of samples (e.g. SSAO) or by voxelizing the scene (e.g. GI with Voxel Cone Tracing).

If the (1) is true, why is it that in SSAO, the sample points are in the space within the hemisphere? Shouldn’t the sample points be located at a random point on the hemisphere’s “surface”?