  The island formula -- an extremization prescription for generalized entropy
-- is known to result in a unitary Page curve for the entropy of Hawking
radiation. This semi-classical entropy formula has been derived for
Jackiw-Teitelboim (JT) gravity coupled to conformal matter using the "replica
trick" to evaluate the Euclidean path integral. Alternatively, for eternal
Anti-de Sitter black holes, we derive the extremization of generalized entropy
from minimizing the microcanonical action of an entanglement wedge. The
on-shell action is minus the entropy and arises in the saddle-point
approximation of the (nonreplicated) microcanonical path integral. When the
black hole is coupled to a bath, islands emerge from maximizing the entropy at
fixed energy, consistent with the island formula. Our method applies to JT
gravity as well as other two-dimensional dilaton gravity theories.
