Research into didactics of calculus has maintained a long-standing interest in students' grasp of the relations between definite integrals and areas. This study comes to contribute to this line of research by unpacking how students use the concept of area to find definite integrals. Specifically, we focus on mathematical situations where the integral functions are represented graphically and change signs between the limits of integration. We conceptualize these situations through the lens of Fischbein's theories of mental models and contemporary modelling literature. This lens is applied to authentic data corpus collected in a large first-year mathematics course--video-recordings of solutions that students submitted as part of their coursework and students' responses in multiple-choice final exams. The analysis revealed two modelling paths that students' solutions followed: one based on areas that the function enclosed, and one based on the integrals of the assigned functions. While the key role of areas in the first path could be expected, area-based reasoning in the second path is noteworthy. In the exams, just above one-fifth of students' responses were consistent with adding the positive measures of the enclosed areas. We situate these findings in the calculus literature to draw educators' attention to silent challenges that are inherent in the integral--area relations.