Prior research on students' productive understandings of definite integrals has reasonably focused on students' meanings associated to components and relationships within the standard definition of a limit of Riemann sums. Our analysis was aimed at identifying (i) the broader range of productive quantitative meanings that students invoke and (ii) the ways in which these meanings interact and evolve throughout successful modeling activity. We conducted a retrospective analysis of data from prior classroom teaching experiments and task-based interviews with 84 students in introductory calculus courses working within 23 small groups on modeling tasks involving definite integrals. We present our results in terms of an Emergent Quantitative Models framework derived from the observation that students reasoned with different quantitative meanings when shifting between the processes and scales involved in the construction of a definite integral. Students' "basic models" are their quantitative interpretation of a relationship in which all of the quantities can be conceived as constant, often represented by well-known formulas, such as [distance] = [velocity]·[time]. For situations in which one or more quantities vary, students' "local models" are their reinterpretation of a basic model applied to a subregion of the phenomenon with limited variation. Students' corresponding "global models" are conceived as an accumulation of these local components. We document the primary versions of these models and provide examples of their nature and co-evolution throughout students' modeling activity and their role in successful completion of the tasks and interpretation of definite integrals. Finally, we present a characterization of an idealized productive "Quantitatively Based Summation" understanding of definite integrals.