Explanation for the integral of differential forms In our course of differential geometry we defined the integral $\int_{U} \omega$ of a differential form $\omega=f dx_1\wedge \ldots \wedge dx_n: T^nU \rightarrow \mathbb R$ with $U\subseteq \mathbb R^n$, $U$ open and compact support $\operatorname{supp}(\omega)$ via $$\int_U \omega := \underbrace{\int_U f(x_1,\ldots,x_n) dx_1 \ldots dx_n}_\text{Riemann integral}$$
This is the same definition Manfredo do Carmo used in his book "Differential Forms and Applications" [chapter 4.1, page 55].
Unfortunately there was no motivation given for this definition in our course. For me as a student it seemed that our professor set $\int_U f dx_1\wedge \ldots \wedge dx_n$ to be $\int_U f(x_1,\ldots,x_n) dx_1 \ldots dx_n$ because both notions are similar. But the reasoning must be vice versa: Because both concepts of integration lead to the same result the notations are similar.
So why does the above definition make sense? Is there some kind of intuitive idea behind the integral of $\omega$ from which I can see that it is the usual Riemann integral? What is the concept/idea behind integrating differential forms? (For example I can think of the Riemann integral as the limit of the area under step functions)

Once I heard differential forms being introduced in a classroom as "the stuff we want to integrate". Take a look at the change of variables formula for integration in $\mathbb{R}^n$ (the Jacobian formula) and you will see it behaves the way differential forms do. The impression one gets is that differential forms were created to simplify integration. I think the motivation is clear if you look at the properties that define a differential form as being tailor made to be used in integration. I  agree the motivation is opaque if you think of forms by themselves and then ask why their integral is defined in that way. Also of course, once a mathematical object is defined, mathematicians will explore its properties independent of the reason that leads to its creation. Complex numbers were created to express solutions of polynomial equations but obviously their use has gone beyond that, with forms is the same story.
