Integrals of Differential Forms I am working out of Munkres Analysis on Manifolds and I see that he claims for $\eta = f dx_1 \wedge \ldots \wedge dx_k$.
$$ \int_A\eta = \int_{x \in A} \eta(x)\big((x;a_1) ,\ldots, (x;a_k) \big)$$
when $a_1,\ldots,a_k$ is a right handed orthonormal basis for $\mathbb{R}^k$.
I believe I understand the definition of each of these objects independently except for the right hand side of the equals. I am not sure how exactly the integral over a k-form can be the same as an integral over a k-tensor, though it seems clear that it has to do with the description of the basis.
I would like a hint as to how to show this or understand these types of integrals better. 
 A: The integral of a $n$-form $\eta$ on a orientable differentiable manifold of dimension $n$ (in the present case, an open submanifold of $\mathbb{R}^n$) is defined in such a way that it agrees with its expressions in coordinates. Precisely, if a coordinate system $(U, \phi=(x_1\ldots x_n))$ on $M$ is given, then on $U$ the $n$-form has a unique expression as
$$\tag{1}\eta= f dx_1\wedge \ldots \wedge dx_n.$$
Then we define
$$\tag{2}\int_U \eta = \int_{\phi(U)} f(t_1\ldots t_n)\, dt_1\ldots dt_n,$$
where the rightmost is the ordinary Riemann integral of multivariable calculus. Turns out that the way $\eta$ changes if we pass to another coordinate system (with the same orientation) is precisely the change of variable formula of multiple integrals, so that (2) is well defined (modulo some technicalities regarding the convergence of the integral,  partitions of unity and the role of orientation). 
Now, to answer your question. If $\eta$ is a $n$-form on an open subset $A$ of $\mathbb{R}^n$, and if $a_1\ldots a_n$ is an orthonormal basis of $\mathbb{R}^n$, then the assignment
$$f(t_1\ldots t_n)=\eta(t_1\ldots t_n)(a_1\ldots a_n)$$
yields exactly the function $f$ we had in equation (1). So, by definition,
$$\int_A \eta = \int_A f(t_1\ldots t_n)\, dt_1\ldots dt_n,$$
which is the formula the book is giving.
