# The microcanonical ensemble approach to calculating the entropy of an ideal gas [duplicate]

I would like to set up the following problem. Assume I have a box of volume $$V$$ with $$N$$ noninteracting particles in it. The energy of each particle can be $$\mathcal{E}_i$$ such that $$\sum_i \mathcal{E}_i = E$$. I want to calculate how many microstates correspond to this macrostate of $$(E, V, N)$$.

Here is my approach so far. I will use the trick where we divide the phase space into boxes of size $$h^3$$. All the particles can be interchanged and we they are distinguishable so we have

$$\Omega(E,V,N) = \frac{1}{N! h^3}\int_V d^N\mathbf{q} \int_{\sum_{i=1}^N p^2_i = 2mE} d^N\mathbf{p}$$.

The second term is where I'm not sure how to proceed. How do I express the momentum integral while respecting the constraint that the total energy is exactly $$E$$?

The larger context: I hope to eventually arrive at some expression for $$\Omega$$ that let's me calculate $$S = k\ln\Omega (E,V,N)$$. My goal is to show that $$\frac{\partial S}{\partial E} = \frac{1}{k_B T}$$ is satisfied and connect to the statistical mechanical definition of temperature.