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In all the notes that I've found on differential geometry, when they introduce integration on manifolds it is always done with top forms with little or no explanation as to why (or any intuition). From what I've manage to gleam from it, one has to use top forms to unambiguously define integration on a manifold (although I'm not quite sure why this is the case?!) and one can integrate lower dimensional forms via integration on a chain (through defining pullbacks). I'm really struggling to understand these notions, please could someone enlighten me on the subject?