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Mar 9, 2019 at 22:14 comment added lcv @user136540 If your problem is how to compute that "integral" numerically, there are certainly better answers. However note that (as Liviu Nicolaescu pointed out) if $G$ is not a "nice function" then your integral is not well defined. This has the practical meaning that if you try to compute that object with said methods you will get answers that will depend on the algorithm used, including errors.
Mar 5, 2019 at 2:29 comment added user136540 Thank you for answering. Unfortunately, G is not a nice function, so that I should calculate it in a numerical way, random samplings for random variable X. I was able to obtain analytical sets satisfying above condition (roots of $G(x)$-$g_c$)
Mar 4, 2019 at 14:09 history answered Jean Duchon CC BY-SA 4.0