Our ability to solve integrals is much more limited when the dx represents a slight change in a function, rather than a small change in a real number. As a result, a lot of things that are easy to say in English such as "quantized curvature in spacetime," or "strongly coupled gauge theory," turn into a big mess when they're written down more precisely. One of the consequences of this limitation is that we have a model for quantized vibrations in spacetime that only works when they do not interact with each other. General relativity says that no, gravitational fields do interact with each other - so the picture we have at present is incomplete. The model of non-self-interacting gravity is a particle we call a "graviton," and it probably describes reality very well when the gravitation involved is so weak that its self-interaction is undetectable.
String theory and loop quantum gravity fit into this picture by trying to replace the integral over something we can't handle with an integral that matches it at large scales, but turns into something more tractable at small scales. Maybe the fact that we still can't make sense of the integral is Nature's way of telling us that she does not do the integral either...