When we specify the internal energy to be U, what we really mean is that the total energy of the gas lies somewhere in an interval of length δ δ U around U. Here δ δ U is taken to be very small, it turns out that the entropy doesn't depend strongly on the choice of δ δ U for large N. This means that the above "area" φ must be extended to a shell of a thickness equal to an uncertainty in momentum δ δ p = δ δ ( 2 m U ) = m 2 U δ δ U , so the entropy is given by: