NOTE ON THE TEACHING OF THE DEFINITE INTEGRAL
The current trends in the Spanish university have forced educators to modify the traditional style of teaching, in the sense that they have to provide a very simplified version of the main concepts of science, especially when it comes to teaching applied mathematics. In general terms, this means that academic discourse must keep away from theoretical formalism. Now the educational message gets even closer to developing intuition and ability to apply science in solving specific problems. Such an approach seems to be dangerous because rigour can be decreased, so the student must use more intensively bibliographical resources.
Among other topics, an important one is the definition of the definite integral. The presentation of the integral of a function is in itself a didactic problem of great interest in teaching elementary maths. Often, it is introduced as the limit of certain finite sums which have been carefully designed. Such a limit, if exists, leads to an acceptable definition of area under a curve. The importance of this approach is well recognized but may not be appropriate in certain contexts as the first course in applied science degree which does not need a rigorous mathematical treatment. As an alternative we propose to start from the primitive notion of area to define the definite integral and then, try to derive the main properties. The advantages and disadvantages of this approach, which have been indirectly examined by Courant and Robbins, are analyzed in this work. They are also described some experiences of the authors in applying this approach.