I´m currently studying these topics, but I think that there is a lack of formalism in the books that I´m reading: Apostol vol 2; Marsden; Thomas vol 2.
So I would really appreciate if you can recommend me books where every detail in this topics is formally justified.
I like Harold Edwards' Advanced Calculus: A Differential Forms Approach. As you may have guessed from the name though, he does take a differential forms approach to line and surface integrals.
Quite a bit of chapter 2 is devoted to formalizing integration of $2$-forms (double integrals/ surface integrals) -- especially section 2.3. He doesn't give every detail -- as then how would you learn? -- but he does go through quite a bit of it.
It is then clear from the discussions throughout chapters 1 and 2 how to generalize this to integration of $1$-forms (line integrals) and $3$-forms (volume integrals) in $\Bbb R^3$. Later in chapter 6 he introduces integration in a more general context (integration on manifolds).
I don't know if it's exactly what you want but it's definitely worth a look.