In Cartan calculus, integrating a $k$-differential form $\alpha$ on a $k$-cubic chain $\sigma$ doesn't need more than the Riemann integral: $$\int_\sigma \alpha := \sum_{i=1}^N c_i \int_{I^k}\sigma_i^*(\alpha) = \sum_{i=1}^N c_i \int_0^1dx_1 \cdots \int_0^1dx_k f(x_1,\ldots,x_k)$$ where the $\sigma_i$ are standard smooth cubes and $$\sigma = \sum_{i=1}^N c_i \sigma_i \mbox{ and } f(x_1,\ldots,x_k) = \sigma_i^*(\alpha)_x(e_1,\ldots,e_k)$$ And with that you can do a lot already if not almost everything: Stokes' theorem, Cartan formulae, variation of integral of forms on chains (variation calculus) etc. (It is BTW what I used to extend Cartan calculus to diffeology, where there is no Lebesgue measure). Of course, differential calculus is not integration theory. But I would not over-teach and use tools where they are far beyond the needs. So the choice depends, as usual, on our goal :)