  We introduce and study a general concept of integral of a threetuple (H, A,
C), where H is a Hopf algebra acting on a coalgebra C and coacting on an
algebra A. In particular, quantum integrals associated to Yetter-Drinfel'd
modules are defined. Let A be an H-bicomodule algebra, $^H {\cal YD}_A$ be the
category of (generalized) Yetter-Drinfel'd modules and $B$ the subalgebra of
coinvariants of the Verma structure of $A$. We introduce the concept of quantum
Galois extensions and we prove the affineness criterion in a quantum version.
