Alternative conceptions of university students regarding the concept of the definite integral

Understanding the concept of definite integral continues to pose a challenge in university mathematics education, as students often develop alternative conceptions that differ from those accepted by the mathematical community. In this context, this article presents the results of a qualitative case study conducted to identify and characterise these conceptions among university students at a mathematics faculty in the state of Guerrero, Mexico. For data collection, a task-based interview was administered to 10 students, and the thematic analysis method was used to analyse the information obtained. In total, six alternative conceptions of the integral were identified, which are as follows: (1) the integral interpreted as positive area; (2) the derivative and the integral as inverse operations; (3) the indefinite and definite integrals seen as disconnected concepts; (4) the graph of the integral conceived as a graphical transformation of the derivative; (5) the sign of the area under the curve established from the sign of the x-axis on which the graph of the function lies; and (6) the integration constant interpreted as area. These findings invite reflection on the need for future research to promote apprehension of the concept of the integral among university students who will eventually enter the field of mathematics teaching.