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In the paper, the quantum integrals associated to quantum Yetter-Drinfeld  π-modules are defined. We shall prove the following affineness criterion: if  there exists θ = {θβ : Hβ → Hom(Hβ-1, A)} β   a total quantum integral and  the canonical map χ : AB A →  H  A, χ(aB b)=  S-1 (b[1,-1-1])b[0,0]  ab[0,0] is subjective. Then the  induction functor -B A : UB → H YDA is an equivalence of categories. The  affineness criterion proven by Menini and Militaru is recovered as special  cases.

Quantum integrals and the affineness criterion for quantum Yetter-Drinfeld π-modules