Just a modest proposal so a comment (but I am not entitled). Every Riemann (even Lebesgue) integrable function, say on the line, can be regarded in a natural way as a distribution (the distributional derivative of its primitive which is continuous and so itself a distribution). There is a simple and elementary (level of a freshman course in univariate analysis) theory of definite integrals for distributions which then provides an answer to your query (not necessarily the one you are looking for). In case of interest, I would be happy to provide details and references.
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