What is the relation of $\int f dx^1\wedge dx^2\wedge ...\wedge dx^n=\int f dx^1...dx^n$

Question

In a book "calculus on manifolds" it is defined that $\int f dx^1\wedge dx^2\wedge ...\wedge dx^n=\int f dx^1...dx^n$ but how it is possible the relate the integrand of a multilinear function (n-differential form) with the remann integral. when i am learning remann integration, i considered $dx$ as an infinitestimal distance although the lecturer didn't explain isolately what $dx$ is. I don't quite see the relationship between remann integral and tensors

Answer 1

This is a definition. See bottom of pg. 100 of Spivak's book.

If $\omega$ is a $k$-form on $[0,1]^k$, there is a unique $f$ such that $\omega = f dx^1 \wedge \cdots \wedge dx^k$. Then define $$ \int_{[0,1]^k} \omega := \int_{[0,1]^k} f $$ or $$ \int_{[0,1]^k} f dx^1 \wedge \cdots \wedge dx^k= \int_{[0,1]^k} f(x^1,\cdots,x^k)dx^1\cdots dx^k $$