I am trying to integrating something like:
My issue is that doing the full integral takes too long to compute, so I am trying to approximate it. I tried two approximations: 1) using the analytical form of the integral and substituting in the limits of integration. 2) approximating the integral as a sum and summing up over the x and z indices.
However, both of these methods give me a result that is significantly different from the full integral. The strangest part is that both (1) and (2) give me the same exact (wrong) solution. I think this may have to do with the complex values of the integrand. The absolute value of the solution is what I ultimately need. I wonder if I need to be careful to account for the real and imaginary parts of the integrand when approximating as a sum.
I have done similar integrals before and haven't had issues. What are the possible problems that I could be facing? How is Integrate acting differently than taking the analytical form of the integral and plugging in values, or summing up the integrand over the relevant indices?