# 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 $$(\vec q, \vec p)$$?