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We investigate upper-division student difficulties with direct integration inmultiple contexts involving the calculation of a potential from a continuousdistribution (e.g., mass, charge, or current). Integration is a tool that hasbeen historically studied at several different points in the curriculumincluding introductory and upper-division levels. We build off of these priorstudies and contribute additional data around student difficulties withmulti-variable integration at two new points in the curriculum: middle-divisionclassical mechanics, and upper-division magnetostatics. To facilitatecomparisons across prior studies as well as the current work, we utilize ananalytical framework that focuses on how students activate, construct, execute,and reflect on mathematical tools during physics problem solving (i.e., theACER framework). Using a mixed-methods approach involving coded exam solutionsand student problem-solving interviews, we identify and compare students'difficulties in these two different context and relate them to what has beenfound previously in other levels and contexts. We find that some of theobserved student difficulties were persistent accross all three contexts (e.g.,identifying integration as the appropriate tool, and expressing the differencevector), while other difficulties seemed to fade as students advanced throughthe curriculum (e.g., expressing differential line, area, and volume elements).We also identified new difficulties that appear in different contexts (e.g.,interpreting and expressing the current density).

A cross-context look at upper-division student difficulties with integration