calculation and analysis – Unexpected difference between integral and summation?

I'm trying to incorporate something like: Integrate(Exp(-i*(k*x+k*z))*Exp(-(x^2+z^2)),{x,-largenumber,largenumber},{z,-largenumber,largenumber})

My problem is that doing the full integral takes too much time to calculate, so I try to approach it. I've tried two approximations: 1) using the analytic form of the integral and substituting it within the limits of integration. 2) to approximate the integral as a sum and by summing the indices x and z.

However, these two methods give me a very different result from the complete integral. The strangest part is that (1) and (2) give me exactly the same solution (false). I think that may have to do with the complex values ​​of integrande. The absolute value of the solution is what I ultimately need. I wonder if I have to take care to account for the real and imaginary parts of the integrand when it gets closer to it all.

I have already done similar integrals and I have not had any problems. What are the possible problems that I might face? How does Integrate act differently than taking the analytic form of the integral and plugging values, or summarize the integral on the relevant clues?