Students’ understandings of multivariate integrals and how they may be generalized from single integral conceptions

•Students’ understandings of multiple integrals are explored.•The way these understandings may be generalized from single integral conceptions is explored.•Constructing multiple integral understandings may require significant restructuring to single integral conceptions.•Many multiple integral understandings are based on area under the curve ideas.

Prior research has documented several conceptions students have regarding the definite integral, but these largely describe how students understand single-variable integral expressions. There is little research that has focused specifically on investigating students’ personal understandings of advanced types of integrals, including multiple integrals, nor how these might be generalized from the students’ understanding of single-variable definite integrals. Through interview data from 10 students and survey data from 42 students, we present a variety of student understandings of multiple integrals. We also investigate the relationship between these understandings and students’ conceptions of single-variable integrals.