The performance of integral constitutive equations in the numerical modeling of two simple viscoelastic flows is examined. These flows include one‐dimensional, steady Poiseuille flow, and one‐dimensional, nonhomogeneous, extensional sink flow. The integral constitutive relations examined include the upper‐convected Maxwell, the Wagner, the Doi‐Edwards, and the Curtiss‐Bird models. The numerical solutions, produced via a Galerkin finite‐element procedure and Newton iteration, are compared with analytic solutions for the same flows. It is shown that unrealistic predictions by the constitutive equations, shown by the analytic solution (e.g., nonmonotonic shear or normal stress) result in divergence of the Newton iteration.