• 제목/요약/키워드: definite integral

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An analysis of the introduction and application of definite integral in textbook developed under the 2015-Revised Curriculum (2015 개정 교육과정에 따른 <수학II> 교과서의 정적분의 도입 및 활용 분석)

  • Park, Jin Hee;Park, Mi Sun;Kwon, Oh Nam
    • 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.

A Study on Infinitesimal Interpretation of Definite Integral (정적분의 무한소 해석에 대한 고찰)

  • Joung, Youn-Joon;Kang, Hyun-Young
    • 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.

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An Analysis of the Concept on Mensuration by Parts and Definite Integral (구분구적법과 정적분의 개념 분석)

  • Shin, Bo-Mi
    • 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.

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High School Students' Understanding of Definite Integral (고등학생들의 정적분 개념 이해)

  • Shin, Bo-Mi
    • 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.

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Comprehending the Symbols of Definite Integral and Teaching Strategy (정적분 기호 이해의 특징과 교수학적 전략)

  • Choi, Jeong-Hyun
    • 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.

A study on the Relationship between Indefinite Integral and Definite Integral (부정적분과 정적분의 관계에 관한 고찰)

  • Joung, Youn-Joon;Lee, Kyeong-Hwa
    • 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.

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A study for Build the Concept Image about Natural Logarithm under GeoGebra Environment (GeoGebra 환경에서 정적분을 이용한 자연로그의 개념이미지 형성 학습 개선방안)

  • Lee, Jeong-Gon
    • Journal for History of Mathematics
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    • v.25 no.1
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    • pp.71-88
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    • 2012
  • The purpose of this study is to find the way to build the concept image about natural logarithm and the method is using definite integral in calculus under GeoGebra environment. When the students approach to natural logarithm, need to use dynamic program about the definite integral in calculus. Visible reasoning process through using dynamic program(GeoGebra) is the most important part that make the concept image to students. Also, for understand mathematical concept to students, using GeoGebra environment in dynamic program is not only useful but helpful method of teaching and studying. In this article, about graph of natural logarithm using the definite integral, to explore process of understand and to find special feature under GeoGebra environment. And it was obtained from a survey of undergraduate students of mathmatics. Also, relate to this process, examine an aspect of students, how understand about connection between natural logarithm and the definite integral, definition of natural logarithm and mathematical link of e. As a result, we found that undergraduate students of mathmatics can understand clearly more about the graph of natural logarithm using the definite integral when using GeoGebra environment. Futhermore, in process of handling the dynamic program that provide opportunity that to observe and analysis about process for problem solving and real concept of mathematics.

A LOWER BOUND FOR THE NUMBER OF SQUARES WHOSE SUM REPRESENTS INTEGRAL QUADRATIC FORMS

  • Kim, Myung-Hwan;Oh, Byeong-Kweon
    • Journal of the Korean Mathematical Society
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    • v.33 no.3
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    • pp.651-655
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    • 1996
  • Lagrange's famous Four Square Theorem [L] says that every positive integer can be represented by the sum of four squares. This marvelous theorem was generalized by Mordell [M1] and Ko [K1] as follows : every positive definite integral quadratic form of two, three, four, and five variables is represented by the sum of five, six, seven, and eight squares, respectively. And they tried to extend this to positive definite integral quadratic forms of six or more variables.

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A DEFINITE INTEGRAL FORMULA

  • Choi, Junesang
    • East Asian mathematical journal
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    • v.29 no.5
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    • pp.545-550
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    • 2013
  • A remarkably large number of integral formulas have been investigated and developed. Certain large number of integral formulas are expressed as derivatives of some known functions. Here we choose to recall such a formula to present explicit expressions in terms of Gamma function, Psi function and Polygamma functions. Some simple interesting special cases of our main formulas are also considered. It is also pointed out that the same argument can establish explicit integral formulas for other those expressed in terms of derivatives of some known functions.