Integer partitions are a classic subject having applications across mathematics, with recent interest driven by a connection to geometric complexity theoretic approaches to the P vs NP conjecture. Here we give the first asymptotic formulas and limit shapes for integer partitions of n fitting inside an m x l rectangle in every regime where a limit shape exists. Furthermore, we give an asymptotic generalization of Sylvester's notoriously difficult 1878 unimodality theorem for q-binomial coefficients. Our approach uses a carefully chosen probability distribution on partitions to apply a two-dimensional local central limit theorem. Joint work with Greta Panova and Robin Pemantle.