Abstract: Complete integrability is a condition on a Hamlitonian system in which there are a large number of conserved quantities. An example is the evolution of a spherical pendulum, in which the angular momentum and energy are preserved. Given an integrable system, the loci with fixed values of the conserved quantities are generically Lagrangian submanifolds. In mirror symmetry, the mirroris expected to be a space with a "Landau-Ginzburg potential" given by counting J-holomorphic disks (a la Gromov) bounding fibers. I will talk about some expected properties of the mirror, and discuss some techniques, developed jointly with S. Venugopalan (IMSc Chenna) for tropical computations of the mirror, with the spherical pendulum being an example, and del Pezzo surfaces (with the integrable systems introduced by Vianna) providing other examples.