Prerequisits for Gauss-Green theorem Consider the following theorem from the appendix C from Evans PDE book:




I know about integration in $\mathbb{R}^n$ but not about how to make sense of the integrals on the right-hand side. As my time is very limited I can't read 50+ pages on the general theory integration on manifolds, I am in desperate need of a text that in a minimum amount of space manages to get my aquainted with those integral and the proof of (i) from above.
It's a plus if the text has some worked examples (the proofs from those statements of the text, that are intuitively plausible, I want to skip anyway because I don't have sufficient time).

(Note:This question is a total re-edit. For further details please consult the edit history.)
Perhaps this not-too-long notes will help
http://www.owlnet.rice.edu/~fjones/chap14.pdf

It gives some rather elementary explanations and proof of the Gauss theorem in $\mathbb{R}^n$. For the complete set of notes (used for an honors calculus class at Rice University) covering other related topics, refer to http://www.owlnet.rice.edu/~fjones/
What you're really asking for a textbook giving a modern and careful presentation of vector calculus/calculus of functions of several variables.Since you're reading Evan's text,I'll assume you've had a least a good undergraduate course in real analysis of one variable on metric spaces a la "baby Rudin" or Pugh. I'll also assume you've got a decent background in linear algebra, which is really important in understanding a careful proof of the variants of Stokes' Theorem, which is really what the Guass-Green theorem is a special case of. 

The standard books for learning this material are Calculus On Manifolds by the legendary Micheal Spivak and Analysis on Manifolds by James Munkres. Spivak's book is basically a problem course with quite a few pictures. It's quite rough going,but it's worth the effort if you've got the patience. Munkres is more of a standard textbook and covers the same material with much more detail.Since you're looking for a rapid introduction that'll cover the material in minimum space, Spivak really will fill your needs best if you're willing to work through it.Indeed, the entire point of the book is to build to careful proofs of the variants of Stokes' theorem. Munkres sacrifices brevity for clarity, so if you need more detail, that's a better choice.Be warned,though-Munkres has a bit more in the way of prerequisites then Spivak. Spivak basically assumes only good courses in linear algebra  and rigorous $\epsilon-\delta$ single variable calculus while Munkres assumes in addition, a good working knowledge of basic topology in metric spaces.Both books work mostly in $R^n$ and discuss abstract manifolds at the end in passing. Both are still really good choices-again, it depends on how much time you have to invest. 

A third book you might want to check out for an excellent rigorous discussion  of Green and Stokes' theorems is Loring Tu's An Introduction To Manifolds. The purpose of Tu's book is to give a clear and relatively short introduction to the theory of differential forms and manifolds with only a background in undergraduate algebra and real analysis. Specifically, the purpose of the book is to provide all the essential background in manifolds that's needed to read the classic text on algebraic topology he co-wrote with Raoul Bott, Differential Forms In Algebraic Topology. That being said, the book contains all the basics with many pictures and is extremely readable. Chapters 21-24 contain an in-depth presentation of statements and proofs of both the Green-Gauss theorem and the general Stokes' Theorem, which the GG theorem is a special case of. I think this book will also serve your needs very well.  

I think any of these books will be good choices for you. But again,if brevity is paramount here and you're willing to work through it with a pen and paper, Spivak is probably your best choice.

Good luck!