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?

result1 = Integrate[eq, limits]
limits2 = {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)]};
Integrate[result1, limits2]

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]