Why functions can't be integrated on manifolds

Question

I'm trying to teach myself about differential forms, and my book says that functions can't be integrated on manifolds because the integral isn't coordinate independent, but if the manifold has dimension $n$, you can integrated $n$ forms in a coordinate independent way.

The example given is simple, but I'm struggling with it. when the manifold is just $(\mathbb{R},1_{i.d.})$. Let $x$ and $y=Cx$ be the coordinate, and coordinate change, and $f(x)=1$. Then $\int_a^bf(x)dx=\int_a^bdx=\int_{Ca}^{Cb}\frac{dx}{dy}dy=(b-a)\neq C(b-a)=\int_{Ca}^{Cb} f(y)dy$. So I guess you would say that the Riemann integral is not coordinate invariant. But the claim is that you can integrate one-forms on a one dimensional manifold in a coordinate independent way. But $f(x)dx$ is a one-form, so I'm a little confused.

Answer 1

When considering coordinate changes, one-forms and functions are different things, as you transform them differently when changing coordinates. The Riemann integral of a one-form does not change after a coordinate change in this context, which is what allows classical "u-substitution", but it does change if you try to transform the one-form $f(x)\,dx$ as the function $f(x)$.

Explicitly, the one-form $f(x)\,dx$ should be pulled back along the transformation $x=a(u)$ as $f(x)\,dx \mapsto f(a(u)) a'(u)\, du$, in correspondence with the chain rule, rather than simply $f(a(u)) du$.