# What does integrating the probability density function over all phase space gives us?

For a system of N-3D particles, we have 6N D.O.F and therefore a 6N dimensional phase space. I know that one point in phase space represents a possible state of the system. I also understand that a tiny volume element $$c\cdot dq^{3n}dp^{3n}$$ contains a certain amount of points, which physically represent all the system's with energy between $$E$$ and $$E + \Delta E$$. Also :

$$\rho (q^{3N}, p^{3N},t) \cdot dq^{3n}dp^{3n}$$ represent the probability that the system can be in this volume, in other words is the probability that the system can have an energy between $$E$$ and $$E + \Delta E$$. This much I can understand, but I some uncertainties:

1. Sometime, for simplistic cases or just generalized cases, we see examples in which, a certain shape (a volume) is drawn in the phase space. What exactly does this volume or drawn shape represents, which contains a certain amount of points inside and there are also points outside of it?

2. What does it mean when we integrate the density all over the phase space? What would be the boundaries, if we can do such a thing?

3. We interpret $$\rho$$ as the probability density, which integrated over a volume, gives us the probability that the system is in this region. At the same time I also have seen cases where $$\rho$$ is interpreted as the nr. of points per unit volume in phase space, and consequently it's integration over a volume gives us the nr. of points within this volume. So one interpretation gives us the probability and the other the nr. of states in the volume. Is the interpretation of it as probabilty density, the normalized version of the density of states?