• Title/Summary/Keyword: Indefinite Integral

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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 Case Study on the Relationship between Indefinite Integral and Definite Integral according to the AiC Perspective (AiC 관점에 따른 부정적분과 정적분 관계 학습사례 연구)

  • Park, Minkyu;Lee, Kyeong-Hwa
    • 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.

A NOTE ON SEMI-SLANT LIGHTLIKE SUBMANIFOLDS OF AN INDEFINITE KENMOTSU MANIFOLD

  • Kaur, Ramandeep;Shanker, Gauree;Yadav, Ankit;Ali, Akram
    • Honam Mathematical Journal
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    • v.43 no.1
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    • pp.152-166
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    • 2021
  • In this paper, we study the geometry of semi-slant lightlike submanifolds of an indefinite Kenmotsu manifold. The integrability conditions of distributions D1 ⊕ {V}, D2 ⊕ {V} and RadTM on semi-slant lightlike submanifolds of an indefinite Kenmotsu manifold are defined. Furthermore, we derive necessary and sufficient conditions for the above distributions to have totally geodesic foliations.

A Study of Students' Perception and Expression on the Constant of Distance Function in the Relationship between Distance Function and Speed Function (거리함수와 속력함수의 관계에서 거리함수의 상수항에 대한 학생들의 인식과 표현)

  • Lee, Dong Gun
    • The Mathematical Education
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    • v.56 no.4
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    • pp.387-405
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    • 2017
  • The purpose of this study is to investigate the change of students 'perception and expression about the motion of object following distance function $={x \atop 3}$ and distance function $y=\frac{x^3}{3}+3$ according to the necessity of research on students' perception and expression about integral constant. In this paper, we present the recognition and the expression of the difference of the constant in the relationship between the distance function and the speed function of the students, while examining the process of constructing the speed function and the inverse process of the distance function. This provides implications for the relationship between the derivative and the indefinite integral corresponding to the inverse process. In particular, in a teaching experiment, a constructive activity was performed to analyze the motion of two distance functions, where the student had a difference of the constant term. At this time, the students used the expression 'starting point' for the constants in the distance function, and the motion was interpreted by using the meaning. This can be seen as a unique 'students' mathematics' in the process of analyzing the motion of objects. These scenes, in introducing the notion of the relation between differential and indefinite integral, it is beyond the comprehension of the integral constant as a computational procedure, so that the learner can understand the meaning of the integral constant in relation to the motion of the object. It is expected that it will be a meaningful basic research on the relationship between differential and integral.

INDEFINITE STOCHASTIC OPTIMAL LQR CONTROL WITH CROSS TERM UNDER IQ CONSTRAINTS

  • Luo, Cheng-Xin;Feng, En-Min
    • Journal of applied mathematics & informatics
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    • v.15 no.1_2
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    • pp.185-200
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    • 2004
  • A stochastic optimal LQR control problem under some integral quadratic (IQ) constraints is studied, with cross terms in both the cost and the constraint functionals, allowing all the control weighting matrices being indefinite. Sufficient conditions for the well-posedness of this problem are given. When these conditions are satisfied, the optimal control is explicitly derived via dual theory.

SOME INTEGRAL REPRESENTATIONS OF THE CLAUSEN FUNCTION Cl2(x) AND THE CATALAN CONSTANT G

  • Choi, Junesang
    • East Asian mathematical journal
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    • v.32 no.1
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    • pp.43-46
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    • 2016
  • The Clausen function $Cl_2$(x) arises in several applications. A large number of indefinite integrals of logarithmic or trigonometric functions can be expressed in closed form in terms of $Cl_2$(x). Very recently, Choi and Srivatava [3] and Choi [1] investigated certain integral formulas associated with $Cl_2$(x). In this sequel, we present an interesting new definite integral formula for the Clausen function $Cl_2$(x) by using a known relationship between the Clausen function $Cl_2$(x) and the generalized Zeta function ${\zeta}$(s, a). Also an interesting integral representation for the Catalan constant G is considered as one of two special cases of our main result.

ON INDEFINITE LOCALLY CONFORMAL COSYMPLECTIC MANIFOLDS

  • Massamba, Fortune;Mavambou, Ange Maloko;Ssekajja, Samuel
    • Communications of the Korean Mathematical Society
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    • v.32 no.3
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    • pp.725-743
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    • 2017
  • We prove that there exist foliations whose leaves are the maximal integral null manifolds immersed as submanifolds of indefinite locally conformal cosymplectic manifolds. Necessary and sufficient conditions for such leaves to be screen conformal, as well as possessing integrable distributions are given. Using Newton transformations, we show that any compact ascreen null leaf with a symmetric Ricci tensor admits a totally geodesic screen distribution. Supporting examples are also obtained.

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 the Line impedance calculation method in electrified railway system (전기철도에서 급전선로의 line impedance 계산에 관한 연구)

  • Lee, Chun-Bae;Lee, Jong-Woo
    • Proceedings of the KSR Conference
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    • 2004.06a
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    • pp.1308-1312
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    • 2004
  • Transmission line impedance calculation has been tried for obtaining exact value. The method proposed by Carson contains indefinite complex integral. Although the Carson solution is proposed with power series, the solution is limited and valid at special range of frequency. In this paper, we proposed a simplified Carson solution by analytical method using ground transmission line return current. This method calculate the transmission line impedance easily.

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