• The Mathematical Education
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• v.57 no.2
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• pp.157-177
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• 2018

The students in secondary schools have been taught calculus as an important subject in mathematics. The order of chapters-the limit of a sequence followed by limit of a function, and differentiation and integration- is because the limit of a function and the limit of a sequence are required as prerequisites of differentiation and integration. Specifically, the limit of a sequence is used to define definite integral as the limit of the Riemann Sum. However, many researchers identified that students had difficulty in understanding the concept of definite integral defined as the limit of the Riemann Sum. Consequently, they suggested alternative ways to introduce definite integral. Based on these researches, the definition of definite integral in the 2015-Revised Curriculum is not a concept of the limit of the Riemann Sum, which was the definition of definite integral in the previous curriculum, but "F(b)-F(a)" for an indefinite integral F(x) of a function f(x) and real numbers a and b. This change gives rise to differences among ways of introducing definite integral and explaining the relationship between definite integral and area in each textbook. As a result of this study, we have identified that there are a variety of ways of introducing definite integral in each textbook and that ways of explaining the relationship between definite integral and area are affected by ways of introducing definite integral. We expect that this change can reduce the difficulties students face when learning the concept of definite integral.

• School Mathematics
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• v.10 no.3
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• pp.375-399
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• 2008

Infinitesimal did not play an explicit role concerning definite integral in the textbook nowadays. But studies which investigate understanding of students on definite integral show that many students comprehend definite integral with infinitesimal. Formally infinitesimal is not taught at mathematics classroom, but many students identify definite integral as infinite sum of infinitesimals. This means that definite integral itself contains some structural elements that allow infinitesimal interpretation. In this study we investigate the role of infinitesimal In the historical development of partition-sum in definite integral, extract didactical issues concerning understanding of definite integral, and analyse Korean mathematics textbooks. Finally we propose some suggestions on the teaching of definite integral which contains the process of refinement intuition.

• School Mathematics
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• v.11 no.2
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• pp.301-316
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• 2009

There are two distinct processes, definite integral and indefinite integral, in the integral calculus. And the term 'integral' has two meanings. Most students regard indefinite integrals as definite integrals with indefinite interval. One possible reason is that calculus textbooks do not concern the meaning in the relationship between definite integral and indefinite integral. In this paper we investigated the historical development of concepts of definite integral and indefinite integral, and the relationship between the two. We have drawn pedagogical implication from the result of analysis.

• Journal of the Korean School Mathematics Society
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• v.11 no.3
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• pp.421-438
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• 2008

Understanding the concept of definite integral is based on understanding the concept of mensuration by parts. However, several previous studies pointed out the difficulty on teaching the concept of mensuration by parts. The paper provides some didactic strategies which help teaching the concept of mensuration by part. To teach the concept of definite integral, in the high school curriculum, the relation between definite integral and series is dealt with. However, the paper suggests that importing the concept of series is not indispensable to teach the concept of definite integral. It is proper that definite integral is taught as limit of particular sequence not series.

• Communications of Mathematical Education
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• v.36 no.1
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• pp.39-57
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• 2022

This study aims to design an integral instruction method that follows the Abstraction in Context (AiC) framework proposed by Hershkowitz, Schwarz, and Dreyfus to help students in acquiring in-depth understanding of the relationship between indefinite integrals and definite integrals and to analyze how the students' understanding improved as a result. To this end, we implemented lessons according to the integral instruction method designed for eight 11th grade students in a science high school. We recorded and analyzed data from graded student worksheets and transcripts of classroom recordings. Results show that students comprehend three knowledge elements regarding relationship between indefinite integral and definite integral: the instantaneous rate of change of accumulation function, the calculation of a definite integral through an indefinite integral, and The determination of indefinite integral by the accumulation function. The findings suggest that the AiC framework is useful for designing didactical activities for conceptual learning, and the accumulation function can serve as a basis for teaching the three knowledge elements regarding relationship between indefinite integral and definite integral.

• Journal for History of Mathematics
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• v.24 no.3
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• pp.77-94
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• 2011

This study aims to provide a teaching strategy accommodating the symbols of the definite integral and guiding students through the meaning of notations in area and volume calculations, based on characterization as to how students comprehend the symbols used in the Riemann sum formula and the definite integral, and their interrelationship. A survey was conducted on 70 high school students regarding the historical background of integral symbols and the textbook contents designated for the definite integral. In the following analysis, the comprehension was qualified by 5 levels; students in higher levels of comprehension demonstrated closer relation to the history of integral notations. A teaching strategy was developed accordingly, which suggested more desirable student understanding on the concept of definite integral symbols in area and volume calculations.

• Communications of Mathematical Education
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• v.38 no.3
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• pp.443-458
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• 2024

This study analyzed the mathematical academic perspective and the actual status of the school field on the introduction of a definite integral as a 'Fundamental Theorem of Calculus' in the 2015 revised mathematics curriculum. Therefore, in order to investigate the mathematical academic perspective and the actual status of the school field, a study was conducted with 12 professors majoring in mathematical analysis and 36 teachers. From a mathematical academic point of view, professors majoring in mathematical analysis said that introducing a definite integral as a 'Fundamental Theorem of Calculus' in the 2015 revised mathematics curriculum was difficult to significantly represent the essence and meaning of the definite integral. In addition, in the actual status of the school field, teachers recognize the need for a relationship between a definite integral and the area of a figure, but when a definite integral is introduced as a 'Fundamental Theorem of Calculus', students find it difficult to recognize the relationship between the definite integral and the area of a figure. As the 2022 revised curriculum, which will be implemented later, introduces definite integrals as a 'Fundamental Theorem of Calculus' this study can consider implications for the introduction and guidance of static integrals. And, this study proposed a follow-up study on an effective teaching and learning method that can relate the definite integral to the area of the figure when introducing the definite integral as the 'Fundamental Theorem of Calculus' and on various visual tools and media.

• School Mathematics
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• v.11 no.1
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• pp.93-110
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• 2009

This paper provides an analysis of a survey on high school students' understanding of definite integral. The purposes of this survey were to identify high school students' private concept definitiones and concept images on definite integral. Definitions and images, as well as the relation between them of the definite integral concept, were examined in 108 high school students. A questionnaire was designed to explore the cognitive schemes for the definite integral concept that evoked by the students. The students' individual answers were collected through written environment. Four types of the private concept definitiones and concept images were identified in the analysis.

• Communications of the Korean Mathematical Society
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• v.17 no.1
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• pp.15-20
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• 2002

We present an explicit form for a class of definite integrals whose special cases include some definite integrals evaluated, over a century ago, by Cahen who made use of an appropriate contour integral for the integrand of a well-known integral representation of the Riemann Zeta function given in (3). Furthermore another analogous class of definite integral formulas and some identities involving Riemann Zeta function and Euler numbers En are also obtained as by-products.

• Journal of Ocean Engineering and Technology
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• v.18 no.1
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• pp.7-9
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• 2004

Using the solution to th contour integral of the complex logarithmic function ${\oint}_cIn(z-z_{0})dz$, the following definite integral, derived from the formula to calculate the forces exerted to n circular cylinder by the discrete vortices shed from it, has been evaluated (equation omitted)