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It seems like a monte carlo integration method is doable but pretty difficult to pull off. So I am gonna back away from that and do the integration using while loops and all.
I think i have the correct set up but the answer i get is off by .03. I don't know if i am doing something wrong and getting lucky or it can be that much different. I'll put below an image of the back panel.
I think the disparity is from including the last point or not. It must be an artifact of a double being slightly higher than 1 because if i say stop if x or y gets greater than or equal to 1.01 instead of just 1 (with a step size of .01) i get an answer of 1 which is what i was looking for. Strange I haven't seen this before though. Maybe doing a second dimension exasperates the problem. I am not sure but i'll have to come back to this and double check it later.
i don't trust this yet, but there has to be a normalized airy disk equation out there. so if i can find it i can double check my normalization. it is also possibly my normalized airy disk is so small because my intensity is so high. 125 as can be seen on the left may be a tad unreasonable and something like 1 may better. Same with the prefactor for the airy disk. I chose something so i can see it but it might be a lot higher.