Undergrad Is My Understanding of the Generalized Stokes' Theorem Correct?

[davidge]

I was interested in finding a motivation for the generalized Stokes' theorem. I was asking myself why
$$ \int_D d \omega = \int_{\partial D} \omega $$ for a region $D$ and a (p-1)-dimensional form $\omega$.
Then I found something funny when working with some arbitrary functions in two dimensions. Below is what I found.
In the second image, I called $x^{\mu}_{i}$ and $x^{\mu}_{f}$ the coordinates of the initial and final points, respectively.
My question is: Is this actually correct?

 

[davidge]

I'm not sure about that generalization from 2 to p dimensions that I did above.