I don't have DoCarmo's book on my desk, but in Thorpe's "Elementary topics in differential geometry" it is explained on page 177 what an $n$-surface with boundary is. It is the object $M$ you get when you take the zero-set $S:=f^{-1}(0)$ of a function $f:\ {\mathbb R}^{n+1}\to{\mathbb R}$, cut $S$ along disjoint smooth boundary curves and throw away the unwanted part of $S$ (plus some technical conditions).
It seems that by a "region without boundary" DoCarmo means such an $M$ where no cutting has been done, such that in formulas where there appear integrals along the boundary curves these integrals are trivially zero. A priori this has nothing to do with compactness: The $(x,y)$-plane in $(x,y,z)$-space is such an $M$.