I may be misunderstanding something, but I think the second term (with the double sum in i and j) should be multiplied by 2. Either that or the sum in j should be over j \neq i rather than j<i.
Anyway, you are correct to say that the blue term needs to be expanded further. Just try writing...
P(X>10) = P(X<0) in this specific case. It is due to the fact that the normal distribution is symmetric about the mean. Therefore, P(X> \mu + \epsilon) = P(X< \mu - \epsilon ) for arbitrary real \epsilon. In this case, P(X>10) = P(X> \mu + 5) = P(X < \mu - 5) = P(X < 0).
Correct.
P(X>10) = P(X<0) in this specific case. It is due to the fact that the normal distribution is symmetric about the mean. Therefore, P(X> \mu + \epsilon) = P(X< \mu - \epsilon ) for arbitrary real \epsilon. In this case, P(X>10) = P(X> \mu + 5) = P(X < \mu - 5) = P(X < 0).
The mistake happens in the third line when you convert to polar coordinates. You are no longer integrating over the same region. In the Cartesian case (second line) the integration region is a semi-infinite square with top-right corner at coordinates (0.5,0.5). In the polar case (third line) the...