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The purpose of this article is to demonstrate that i) the framework of elliptic hypergeometric integrals (EHIs) can be extended by input from supersymmetric gauge theory, and ii) analyzing the hyperbolic limit of the EHIs in the extended framework leads to a rich structure containing sharp mathematical problems of interest to supersymmetric quantum field theorists. Both of the above items have already been discussed in the theoretical physics literature. Item i was demonstrated by Dolan and Osborn in 2008. Item ii was discussed in the present author's Ph.D. Thesis in 2016, wherein crucial elements were borrowed from the 2006 work of Rains on the hyperbolic limit of certain classes of EHIs. This article contains a concise review of these developments, along with minor refinements and clarifying remarks, written mainly for mathematicians interested in EHIs. In particular, we work with a representation-theoretic definition of a supersymmetric gauge theory, so that readers without any background in gauge theory - but familiar with the representation theory of semi-simple Lie algebras - can follow the discussion., Comment: Review of arXiv:1512.03376 and arXiv:1605.06100, along with minor refinements and clarifying remarks, written for SIGMA's Special Issue on Elliptic Hypergeometric Functions and Their Applications