We establish the asymptotic expansion of certain integrals of classes on moduli spaces of curves , when either the or goes to infinity. Our main tools are cut-join type recursion formulae from the WittenKontsevich theorem, as well as asymptotics of solutions to the first Painlev equation. We also raise a conjecture on large genus asymptotics for -point functions of classes and partially verify the positivity of coefficients in generalized Mirzakhanis formula of higher WeilPetersson volumes.