Let be a field and let be a finite quiver. We study the structure of the finitely presented modules of finite length over the Leavitt path algebra and show its close relationship with the finite-dimensional representations of the inverse quiver of , as well as with the class of finitely generated -modules such that for all , where is the usual path algebra of . By using these results we compute the higher -theory of the von Neumann regular algebra , where is the set of all square matrices over which are sent to invertible matrices by the augmentation map .