[spunkets]

The integrals always account for some physical interaction proposed in the theory. If numerical methods are used, they are simply finding a numerical solution to the integrals.

[UberVernunft]

If numerical methods are used, they are simply finding a numerical solution to the integrals.

But the integrals are derived from a theoretical set of equations. You could develop a strictly numerical approach based on the original equations -- thus integrals would never be utilized. The integrals are typically derived from a perturbation expansion of the original *physical* equations.

[UberVernunft]

The integrals always account for some physical interaction proposed in the theory.

Huh? You've got it backwards. There are no physical interactions "proposed in the theory". The so called physical interactions are a shorthand picture language for describing the integrals.