In this particular case you don't have to choose between Monte Carlo and using a normal sum approximation. You can just do a non-stochastic numerical integratio

In this particular case you don't have to choose between Monte Carlo and using a normal sum approximation. You can just do a non-stochastic numerical integration (e.g. simple Riemann integration or whatever other method you want to use), since it's only a double integral, and thus the dimensionality isn't so high that you need to resort to stochastic Monte Carlo.