Density distribution of gas in a centrifugal field

The problem asks me to find the density of gas in a cylinder of radius $R$ and length $l$ rotating about its axis with angular velocity $ω$, there being a total of $N$ molecules in the cylinder.

What I have done is shown as following:

I choose to look this scenario in a static frame with the cylinder, which is rotating at an unchanging angular speed $ω$. Then we can consider the field is a centrifugal field.

Thus the energy of a particle can be written as

$$E = E_0 - \frac{1}{2}m\omega^2 r^2\tag{1}$$

I haven't determined $E_0$ yet as I'm not sure about the thermal energy of the particle and the energy caused by the centrifugal field differs with the choosing of zero potential point.

After that, I wrote the probability as

$$\mathrm{d}P = \frac{\rho \mathrm{d}V}{N} = \frac{\rho \cdot 2\pi r\mathrm{d}r\cdot l}{N}\tag{2}$$

where $\rho \mathrm{d}V$ is what I want to find.

My goal is to convert all $r$ and $\mathrm{d}r$ into $E$ and $\mathrm{d}E$ and finally compare it with Boltzmann distribution

$$P(E)=C\cdot e^{-E/kT}\tag{3}$$

where $C$ is a constant.

However, I have met with some problems doing this.

From (1), it can be derived

$$dE=-m\omega^2 r dr\tag{4}$$

After I plug this into (2), I find $P(E)$ in $\mathrm{d}P=P(E)\mathrm{d}E$ is always negative. Thus is cannot be compared with the Boltzmann distribution.

How can I fix this?