In order to derive a simple one-dimensional approach that could handle fluid flows in smooth ducts as well as in ducts of discontinuous cross-section, we propose herein a Finite Volume approach that relies on an integral formulation of the multidimensional flow model. While focusing on Euler equations, we compare two-dimensional results with approximations obtained using the present approach, and also with the classical formulation for variable cross-sections using a well-balanced scheme. Numerical simulations confirm the ability of this integral method to provide approximations that compare well with 2D results. This method also enables to deal with all-even including vanishing-cross-section ducts. This approach may also be applied when considering other single-phase or multi-phase fluid flow models.