Problem with microstates and corresponding volume in the phase space

I’ve been struggling without any result on a formal problem about the relation between microscopic states of a system and the volume occupied by a macrostate in the phase space. I’m now very confused about everything.

I observed that:

• In general entropy is defined as $S = k_b log(Omega(E))$ where $Omega(E)$ is the number of microstates that correspond to a given macrostate of energy E.
• In the microcanonical ensemble a macrostate’s corresponding entropy can be calculated as $S = k_b log(Gamma(E))$ where $Gamma(E)$ is the volume occupied by the macrostate in the phase space (i.e. the set of all possible $vec q$ and $vec p$ that satisfy $H(vec q, vec p, t) = E$.

Why can I use the volume instead of the number of microstates? How are $Gamma(E)$ and $Omega(E)$ related?

For example for an ideal gas in a box the condition that determines the volume in the phase space for a given macrostate of energy E (I’m looking only to the $p$s’ volume) is
$$V = sum_{i = 1}^{3N} frac{p_i^2}{2m} = E$$
which identifies a hypersphere of radius $sqrt{2mE}$.
There are infinite combinations of $vec q$ and $vec p$ that can satisfy the above equation so I would say that there are infinite microstates. Instead the volume is of course finite. How can I connect them?

Concluding, I read something about representative points and about a density function of these representative points but I couldn’t understand well.
What exactly are those representative points and how do they differ from "normal" points in the phase space like $(vec q, vec p)$?

Thank you in advance!