I was trying to calculate the probability of finding a particle with momentum $p$ in the microcanonical ensemble in a 3-dimensional box.
$$\rho_i(\vec p)= \langle\delta(\vec p - \vec p_i)\rangle= \int \mathrm d^3q_1\ldots \mathrm d^3q_N\int \mathrm d^3 p_1\ldots \mathrm d^3p_N \;\rho(\vec q_1,\ldots,\vec p_1,\ldots,\vec p_i,\ldots,\vec p_N)\delta(\vec p-\vec p_i). $$
In the microcanonical ensemble, the phase space density is constant, so our work is to evaluate the last integral over the momentum space. Does the second integral correspond to a volume of a 3 dimensional "hypersphere" or a $3N$ dimensional hypersphere of radius $p$?