We investigate the evolution of the area of multi-dimensional surfaces along the flow of a dynamical system with known first integrals, and we formulate sufficient conditions for area contraction. These results, together with known results about the Hausdorff dimension and the box-counting dimension of invariant sets, are applied to systems exhibiting almost global convergence/asymptotic stability. This leads to a generalization of a well-known result on almost global convergence of a system, based on the use of density functions. We conclude with an example.