On pg. 402 of Lee's Introduction to Smooth Manifolds (Second Edition), the following is said to define the integral of a differential form on $\mathbf R^n$:
Let $D$ be an open domain of integration (a bounded set with its boundary having measure $0$) and $\omega$ be an $n$-form on $\bar D$. Write $\omega=f dx^1\wedge \cdots\wedge dx^n$ for some continuous function $f:\bar D\to \mathbf R$. The integral of $\omega$ over $D$ is defined as $$\int_{D}\omega=\int_Df$$
I am not sure why has the author used $\bar D$ (closure of $D$) above. Is there something wrong with the definition if we replace all instances of "$\bar D$" with "$D$"?