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I have the following 2D Gaussian:

{m[r_, t_], Sigma[S11_,S22_,p_]} := {{r Cos[t],r Sin[t]},{{S11,p S11 S22},{p S11 S22,S22}}};
h[r_, t_, S11_, S22_, p_] := PDF[MultinormalDistribution[m[r, t], Sigma[S11, S22, p]], {x, y}]

I want to integrate this 2D Gaussian of the x y region between the lines y = Tan[\[Pi]/8] x and y = Tan[3\[Pi]/8] x. Specifically, I want to find:

Integrate[h[r, \[Pi]/8, s, s, 0], , {x, 0, Infinity}, {y, Tan[\[Pi]/8] x, Tan[3 \[Pi]/8]  x}];

How can I recast the problem so that I get a result from this?

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Likely no chance. Even Integrate[ h[1, \ [Pi]/8, 2, 2, 0], {x, 0, Infinity}, {y, Tan[\[Pi]/8] x], Tan[3 \[Pi]/8] x}] performs only the result of the integration by y from Tan[\[Pi]/8] x to Tan[3 \ [Pi]/8] x . I think the integral under consideration can be taken only numerically as a function of the parameters r and s:

f[r_?NumericQ, s_?NumericQ] := NIntegrate[h[r, \[Pi]/8, s, s, 0], {x, 0, Infinity}, 
{y, Tan[\[Pi]/8] x, Tan[3 \[Pi]/8] x}]
Plot3D[f[r, s], {r, 0, 2}, {s, 0.01, 2}]

enter image description here

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