Integrative Calculus Module
Albert Q Jimenez
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Abstract
The basic to the concept of definite integral is area. So far in our study of Mathematics, we solved areas of plane figures like of that of triangles, rectangles and squares, circles, etc. On the previous lesson on calculus, one problem that led to the study of calculus was that of finding the area of any shape. It treated us to solve areas bounded by graphs of certain functions in the Cartesian plane. This led us to the concept of integral calculus. Now, in this section, you will learn how to reverse the process of differentiation using the properties of definite integrals. An anti-derivative of a function f(x) is a function f(x) such that F'(x) = f(x) for all x. Objectives 1. Define what anti-derivative is. 2. Determine the uses of signs and notation of integrals. 3. Compute the anti-derivative of a function. Recall The Derivative The first derivative of a function at a point is the slope of the tangent to the curve of the function at that point. The concept is defines precisely as follows. The slope m of a straight line is defined as-The ratio of the rise (change in vertical distance, Δy) to the run (change in horizontal distance, Δx). m = = =-The tangent of its angle of inclination. m = tan θ =
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Bulletin of Symbolic Logic, 1995
This is not about the symbolic manipulation of functions so popular these days. Rather it is about the more abstract, but infinitely less practical, problem of the primitive. Simply stated:Given a derivative f: ℝ → ℝ, how can we recover its primitive?The roots of this problem go back to the beginnings of calculus and it is even sometimes called “Newton's problem”. Historically, it has played a major role in the development of the theory of the integral. For example, it was…
The antiderivative understanding by students in the first university courses
2017
In this article, we present the results of a questionnaire designed to evaluate college students’ understanding of the antiderivative. Specifically, by civil engineering students when answering the questionnaire’ tasks, in order to identify and characterize the meanings on the antiderivative that are mobilized by them. In order to analyse the answers given, we used some theoretical and methodological notions provided by the theoretical model known as the Onto-Semiotic Approach (OSA) of mathematics cognition and instruction. The results show knowledge of antiderivative by the Civil Engineering students. Furthermore, the comparison between the mathematical activity of students provides information that allows concluding that the meanings that they mobilized might be shared among their communities.
Meanings that engineering students confer to the antiderivative
Visión Electrónica, 2019
In this article, are presented the results of a questionnaire designed to evaluate college students' understanding of the antiderivative. Specifically, by civil engineering students when answering the questionnaire tasks, in order to identify and characterize the meanings on the antiderivative that are developed by them. In order to analyse the answers given, were used some theoretical and methodological notions provided by the theoretical model known as the Onto-Semiotic Approach (OSA) of mathematics cognition and instruction. The results show the knowledge of antiderivative by the Civil Engineering students. Furthermore, the comparison between the mathematical activity of students provides information that allows concluding that the meanings that they mobilized might be shared among their communities.
Modelling the height of the antiderivative
2009
Two complementary processes involved in mathematical modelling are (1) mathematising a realistic situation, and (2) applying a mathematical technique to a given realistic situation. We present and analyse work from four undergraduate students and two secondary school teachers who engaged in both processes during a mathematical modelling task that required them to find the antiderivative of a function presented graphically. When determining the height of the antiderivative, they mathematised the situation to develop an elementary mathematical method, and attempted to apply some ideas about definite integration that they had previously learned in class. In the end, however, the participants favoured their more elementary mathematised knowledge over the sophisticated knowledge they tried to apply.
Teaching demonstration of the integral calculus
International Journal of Mathematical Education in Science and Technology, 2019
A simple in-class demonstration of integral Calculus for first-time students is described for straightforward whole number area magnitudes, for ease of understanding. Following the Second Fundamental Theorem of the Calculus, macroscopic differences in ordinal values of several integrals, F(x), are compared to the regions of area traced out from the horizontal axis by the derivative functions f (x) over various domains. In addition, microscopic incremental differentials of an integral at a particular position, dF(x), are compared to corresponding values of the derivative function f (x) multiplied by various horizontal shifts, dx. For any area to exist for a derivative function, dx > 0, but the difference between these compared magnitudes collapses to zero as long as dx widths are small. The demonstration readily confirms, both arithmetically and graphically for trigonometric, polynomial, and transcendental functions, the Newton discoveries that (1) the rate that area accumulates under a function is proportional to the ordinal value of the function itself, and (2) changes in elevation along an integral function automatically equal the exact net area traced out by its derivative from the X-axis.
Setting the calculus straight
The study of the calculus occupies the major part of the pure mathematics Alevel. It is full of formulae and procedures for finding derivatives, integrals, solutions of differential equations, and all manner of other horrendous things. But the underlying ideas are extremely simple-so simple that it is only recently that we have realised how simple they are. In this article David Tall shows how new methods of visualizing the ideas of the calculus can set the ideas straight-in a way that makes them easy to imagine.
Misconceptions and errors in learning integral calculus / Voon Li Li, Nor Hazizah Julaihi and Tang Howe Eng
Penerbit UiTM (UiTM Press), 2017
The paper presents the results of a case study examining students' difficulties in the learning of integral calculus. It sought to address the misconceptions and errors that were encountered in the students' work solution. In quantitative study, the marks obtained by 147 students of Diploma in Computer Science in advanced calculus examinations were used as a measurement to evaluate the percentages of errors. Further, qualitative study examined the types of errors performed by 70 diploma students of the advanced calculus courses in their ongoing assessments. The students encountered more difficulties in solving questions related to improper integrals for standard functions (63.1 percentages of errors). The three techniques of integration, namely by parts, trigonometric substitution and partial fraction with combined percentage errors of 42.8 also contributed to this. The types of conceptual errors discovered are symbolic, standard functions recognition, property of integral and technique determination. The procedural errors are due to the confusion between differentiation and integration process while the technical errors have foreseen the students struggling with poor mathematical skills and carelessness. The results will thus be useful to Mathematics educators who are keen in designing functional teaching and learning instruments to rectify the difficulties and misconceptions problems experienced by calculus students.
Definitions and Images for the Definite Integral Concept
1997
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 Definite Integral and Computer
In this paper the results of the learning concept of definite integral and numeric integration with the computer is presented. The tested students attend ˇSabac Chemical Technological college. The aim of this test was to check the student's theoretical, visual and practical knowledge of definite integral. In almost all secondary schools definite integral and its applications are studied. The concept of definite integral is almost always introduced as the Riemann integral, which is in turn defined in terms of Riemann sums, and its geometric interpretation. For secondary school pupils, as well as for high school and university students, this definition is hard to understand. The courses of Numerical mathematics at all levels contain topics on numerical integration, which is partly based on Riemann sums. With the aid of mathematical software for visualization and computation of approximate integrals, the notion of definite integral and its calculation is more easily adopted by pupi...
The Idea of the Integral (page 181) CHAPTER 5 INTEGRALS 5.1 The Idea of the Integral (page 181
Problems 1-3 review sums and differences from Section 1.2. This chapter goes forward to integrals and derivatives. 1. If fo, f l , f2, f3, f4 = 0,2,6,12,20, find the differences vj = f,-fj-and the sum of the v's. 0 The differences are vl, v2, VQ, v4 = 2,4,6,8. The sum is 2 +4 +6 + 8 = 20. This equals f4-fo. 2. If vl, v2, VQ, v4 = 3 , 3 , 3 , 3 and fo = 5, find the f's. Show that f4-fo is the sum of the v's. O E a c h n e w f j is fj-1 + v j. So f i = f o + v l = 5 + 3 = 8. Similarly f2 = fl + v 2 = 8 + 3 = 11. Then f3 = 14 and f4 = 17. The sum of the v's is 12. The difference between fiaSt and ffirSt is also 17-5 = 12. 3. (This is Problem 5.1.5) Show that f j = 5 has differences vj = f j-fj-l = d-1.-The formula gives fo = = Then fl = & and f2 = 2-r-l a * Now find the differences: In general fj-fj-1 = ri-'. This is vi. Adding the v's gives the geometric aerie. l + r + r 2 +. .+rn-l. 1-r-1 Its sum is f,-fo = 5-r-1-r-1 4. Suppose v(x) = 2x for 0 < x < 3 and v(x) = 6 for x > 3. Sketch and find the area from 0 t o x under the graph of v(x). m There are really two cases to think about. If x < 3, the shaded triangle with base x and height 2 s has area = :(base) (height) = kx(2x) = x2. If x > 3 the area is that of a triangle plus a rectangle. The triangle has base 3 and height 6 and area 9. The rectangle has base (x-3) and height 6. Total area = 9 + 6(x-3) = 62-9. The area f (x) has a two-part formula: Areas under v(x) give f(x) small triangle L x (2 x) = x 2 2 complete triangle ' 3 (6) = 9 2 add rectangle 6(x-3) = 6x-18 5. Use four rectangles to approximate the area under the curve y = $x2 from x = 0 to x = 4. Then do the same using eight rectangles. The heights of the four rectangles are f (1) = i, f (2) = 2, f (3) = $, f (4) = 8. The width of each rectangle is one. The sum of the four areas is 1. $ + 1 2 + 1 + 1 8 = 15. The sketch shows that the actual area under the curve is less than 15.
Related papers
Analysis of the Meanings of the Antiderivative Used by Students of the First Engineering Courses
International Journal of Science and Mathematics Education, 2017
In this article, we present the results of the administration of a questionnaire designed to evaluate the understanding that civil engineering students have of the antiderivative. The questionnaire was simultaneously administered to samples of Mexican and Colombian students. For the analysis of the answers, we used some theoretical and methodological notions provided by the theoretical model known as Onto-Semiotic Approach (OSA) to mathematical cognition and instruction. The results revealed the meanings of the antiderivative that are more predominantly used by civil engineering students. Also, the comparison between the mathematical activity of Mexican and Colombian students provides information that allows concluding that the meanings mobilized could be shared among their communities and are not particular of their classroom or university.
The Teaching and Learning of Definite Integrals: A Special Issue Guest Editorial
International Journal of Research in Undergraduate Mathematics Education
The Existence of The Definite Integral: Students’ Understanding
Proceedings of the Proceedings of the 4th Progressive and Fun Education International Conference, Profunedu 2019, 6-8 August 2019, Makassar, Indonesia, 2019
The definite integral is a part of a compulsory subject, Integral Calculus, for mathematics education students. As pre-service mathematics teachers, students' understanding of the definite integral concepts is crucial to note. This article presents the result of the investigation students' understanding of the definition of the definite integral and its existence. The study reported in this paper was done through qualitative research. A questionnaire about the definition of the definite integral and the existence of the definite integral of a defined function on a closed interval was given to 163 students who took Integral Calculus course. Then, 4 of them were selected to interview in depth. The students' responses indicated the diversity of the conceptual understanding of the existence of the definite integral. Various students' understanding of the definition of the definite integral that leads to students' understanding of the existence of the definite integral is presented as well.
DIFFERENCIATION AND INTEGRATION
Differentiation and integration are basic mathematical operations with a wide range of applications in many areas of science. It is therefore important to have good methods to compute and manipulate derivatives and integrals. You probably learnt the basic rules of differentiation and integration in school -symbolic methods suitable for pencil-and-paper calculations. These are important, and most derivatives can be computed this way. Integration however, is different, and most integrals cannot be determined with symbolic methods like the ones you learnt in school.
Misconceptions and Errors in Learning Integral Calculus
2017
The paper presents the results of a case study examining students’ difficulties in the learning of integral calculus. It sought to address the misconceptions and errors that were encountered in the students’ work solution. In quantitative study, the marks obtained by 147 students of Diploma in Computer Science in advanced calculus examinations were used as a measurement to evaluate the percentages of errors. Further, qualitative study examined the types of errors performed by 70 diploma students of the advanced calculus courses in their on-going assessments. The students encountered more difficulties in solving questions related to improper integrals for standard functions (63.1 percentages of errors). The three techniques of integration, namely by parts, trigonometric substitution and partial fraction with combined percentage errors of 42.8 also contributed to this. The types of conceptual errors discovered are symbolic, standard functions recognition, property of integral and tech...
A Framework to Evaluate University Students' Use of Derive Software to Comprehend and Apply the Concept of Definite Integral
This study documents the type of mathematical competence that first year university students develop as a result of working on series of problems that involve definite integral concepts. During the problem solving sessions students had the opportunity of using DERIVE software to solve the tasks, that were specially designed (Camacho, Depool and Santos, 2004), to comprehend and apply definite integral concepts. In this report, we sketch a framework to evaluate the students' mathematical competences that emerge during the sessions and value the importance, for instructors, to think of the curriculum contents in terms of problems or activities that guide the students' learning trajectories within CAS (Computer Algebra Systems) environment. Introduction The definite integral is a relevant content that engineering students need to comprehend and use in diverse contexts during their learning experiences. Thus, it becomes important to characterize initially, mathematical features, ...
CALCULUS -DIFFERENTIAL, INTEGRAL, AND INFINITE
Garry Alfonsius, 2020
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.
StudentsDifficulties with Application of Definite Integration 1
2007
The purpose of this study is to determine what special learning difficulties may be about the practices of definite integral. With this purpose, a test prepared by the researchers and consisting of 8 openended questions has been applied to 64 students from the Science
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