We consider the asymmetric simple exclusion process (ASEP) on the half line and on a bounded interval with sources and sinks. In the full line, Bertini and Giacomin (1997) proved convergence under weakly asymmetric scaling of the models height function to the solution of the Kardar-Parisi-Zhang (KPZ) equation on the line. We show that under weakly asymmetric scaling of the sources and sinks as well, the half line height function converges to the KPZ equation on the positive real line with Neumann boundary condition at zero; and the bounded interval height function converges to the KPZ equation on the unit interval with Neumann boundary conditions on both sides. This result demonstrates how the KPZ equation arises at the triple critical point (maximal current / high density / low density) of the open ASEP. Our result is proved by applying the Ga ̈rtner transform (or microscopic Hopf-Cole transform) to ASEP. Joint work with Ivan Corwin.