I am confused as to what integral you want to be solved. 

You need to write the integral function in a form like 

    (*integrand*)
    eq = 1;
    (*limits*)
    limits = {x, 2 - Sqrt[2], 1};
    
    result = Integrate[eq, limits]

So, I don't understand your question - are you trying to integrate it twice? 

    eq = 1;
    limits = {y, 
       1/2 - 1/2 Sqrt[(-4 + 8 x - 4 x^3 + x^4)/((-2 + x)^2 x^2)], 
       1/2 + 1/2 Sqrt[(-4 + 8 x - 4 x^3 + x^4)/((-2 + x)^2 x^2)]};
    result1 = Integrate[eq, limits]
    limits2 = {x, 2 - Sqrt[2], 1};
    NIntegrate[result1, limits2]