# Why integrate on cubes that's not injective?

Again, this is a conceptual(soft) problem I had while reading Spivak's calculus on manifold. There, to develop the theory of integration, Spivak chose to integrate k-forms on singular cubes. However, as pointed out here, singular cubes can collapse dimension, so it seems theoretically possible that one pulls back a differential form onto a higher dimension cubical region for evaluating the integral, and I wonder when could that ever be useful? Why don't we just add the assumption that singular cubes are injective? Also, without this injectivity I find the picture of chains even less geometrically intuitive...