Definitions and Images for the Definite Integral Concept Shaker Rasslan David Tall Center for Mathematics Education Oranim School of Education, Israel <shaker@macam.ac.il> Mathematics Education Research Centre University of Warwick, UK <david.tall@warwick.ac.uk> Definitions and images, as well as the relation between them of the definite integral concept, were examined in 41 English high school students. A questionnaire was designed to explore the cognitive schemes for the definite integral concept that are evoked by the students. One question aimed to check whether the students knew to define the concept of definite integral. Five others were designed to categorize how students worked with the concept of definite integral and how this related to the definition. The results show that only 7 students out of 41 of our sample knew the definition. All mathematical concepts except the primitive ones have definitions. Many of them are introduced to high school or college students. However, the students do not necessarily use the definition to decide whether a given idea is or is not an example of the concept. In most cases, they decide on the basis of their concept image, that is, all the mental pictures, properties and processes associated with the concept in their mind. (Tall & Vinner, 1981; Rasslan & Vinner, 1997). The concept of the definite integral is a central in the calculus. In many countries, including the UK, it is taught in the last two years of school to students aged approximately 16–18. The students in this study followed a curriculum based on the School Mathematics Project A-level. In the current version of the textbook (SMP, 1997), integration is introduced through activities to estimate the area between a graph and the x-axis using pictures and numerical methods. After this experience the notion of integral is defined as follows (in the form of a description rather than a formal Riemann sum): The symbol fx dx a b () Ú denotes the precise value of the area under the graph of f between x a = and x b = . It is known as the integral of y with respect to x over the interval from a to b. The integral can be found approximately by various numerical methods. Figure 1: The definition of the integral concept (SMP, 1997, p.143). This is followed by ten pages of experience with numerical approximations including functions with positive and negative values before algebraic integration is introduced in the next chapter. Here the student is encouraged to build up the relationship between polynomials and their (definite) integrals, before the fundamental theorem is introduced using local straightness as a visual form of derivative in the following terms: For any differentiable function f, ¢ = - Ú f x dx fb fa a b () () () . Figure 2: The fundamental theorem of calculus (SMP, 1997, p. 169).