Actually, based on the demonstration graph of the integral region of phi shown upstairs, it should be from pi/4 to pi, because the shared point of cot(phi) and sqrt(2)*sin(theta) is pi/4. However, if we base on this integral bounds ( pi/4 < phi < pi), we cannot get the correct result.

The used integral bound of phi upstairs can work. however, it is short of a rigorous mathematical derivation. This is my pending question.

Thanks