Function providing input and integration limits for NIntegrate

I am trying to define some custom function that evaluates the numeric integral of some complicated function. The problem is that I would like to include as an input the integration limits. A MWE follows

f[x_,y_]:= Exp[-2 x^2 y^2] test[x_?NumericQ,IntL_]:= NIntegrate[f[x, y],IntL] myIntL1= {y,-4 x^2,4x^2}; myIntL2= {y,-4 x^4,4x^4}; 

Then, if I evaluate for instance test[3,myIntL1] I get a problem concerning an invalid limit of integration.

Is there a clever way to fix this without defining several functions including the integration limits, such as

    test1[x_?NumericQ,IntL_]:= NIntegrate[f[x, y],{y,-4 x^2,4x^2}]     test2[x_?NumericQ,IntL_]:= NIntegrate[f[x, y],{y,-4 x^4,4x^4}] 

etc?

Of course, here everything looks simple, but in my case all the functions are rather long; some even purely numeric ones. Since I have multiple choices for the integration limits, it would be more practical to avoid defining test1[x_], test2[x_], ..., testN[x_]

Thanks in advance, Pablo