  Two identities in statistical mechanics involving entropy differences (or
ratios of density of states) at constant energy are derived. The first provides
a nontrivial extension of the Jarzynski equality to the microcanonical ensemble
[C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997)], which can be seen as a
``fast-switching'' version of the adiabatic switching method for computing
entropies [M. Watanabe, W. P. Reinhardt, Phys. Rev. Lett. 65, 3301 (1990)]. The
second is a thermodynamic integration formula analogous to a well-known
expression for free energies, and follows after taking the quasistatic limit of
the first. Both identities can be conveniently used in conjunction with a
scaling relation (herein derived) that allows one to extrapolate measurements
taken at a single energy to a wide range of energy values. Practical aspects of
these identities in the context of numerical simulations are discussed.
