Evaluating students' spatial reasoning in setting up triple integrals

Multivariable calculus is a critical component of STEM education, demanding sophisticated spatial reasoning and abstract geometric thinking to solve advanced problems related to multiple integrals – specifically double and triple integrals. However, the transition from single-variable to multivariable calculus presents significant cognitive challenges for students. This study investigates students' understanding and the underlying cognitive obstacles they face when setting up triple integrals to determine the volume of three-dimensional (3D) solids. A qualitative case study was conducted with undergraduate engineering students, whose written solutions were evaluated across four distinct phases: visualization, projection, coordinate selection, and algebraic setup. Utilizing the Action-Process-ObjectSchema (APOS) theoretical framework as a diagnostic lens, the findings reveal that the primary learning obstacle is not a lack of procedural competence, but rather a fundamental deficit in 3D spatial visualization. While many students successfully executed dimension reduction and coordinate selection, they frequently failed to translate textual descriptions into accurate geometric boundaries, resorting instead to mechanical calculations characteristic of the APOS Action stage. To bridge this cognitive gap, the study recommends a pedagogical shift away from rote algorithmic drills toward instructional strategies that build spatial schema, specifically highlighting the integration of dynamic geometry software like GeoGebra and established pedagogical strategies, such as employing spatial metaphors to conceptualize the bounds of a solid.