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CONTENT •
Acknowledge
•
Introduction
◦ ◦ ◦ ◦
Introduction of project Introduction of integration Definition of integration History of integration
•
Objective
•
Problem Solving
◦ Part1 ◦ Part2 ◦ Part3 •
Further Exploration
•
Conclusion
•
Reflection
•
Reference
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Acknowledge First of all, I would like to say Alhamdulillah, for giving me a strength and health to do this project work. Not forgotten to my parents for providing everything such as money, to buy anything that related to this project work and their advice, which is the most needed for this project. Internets, books, computers and all that. They also support me and encouraged me to complete this task so that I would not procrastinate in doing this. Then, I would like to thank my teacher, Encik Roslan Bin Ab Ghani for guding me and my friends throughout this project. We have some difficulties in doing this task, but he taught us patiently until we knew what to do. He tried and tried to teach us until we understand what we supposed to do with the project work. Last but not least, my friends who were doing this project with me and sharing our ideas. They were helpful that when we combined and discussed together, we had this task done.
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INTRODUCTION OF ADDITIONAL MATHEMATICS PROJECT WORK1/2011 The aims of carrying out this project work are to enable students to : a)Apply a)Ap ply math mathemat ematics ics to everyday ever yday situatio situ ations ns and apprecia appre ciate te theimpor theimporta tannce an andd the the be beau auty ty of mathematics in everyday lives b)Improve problem-solving problem-s olving skills, skill s, thinking t hinking skills , reasoning reas oning and mathe thematica ical communication c)Devel c)De velop op positive posi tive atti attitude tude and personal pers onaliti ities es and instri ins trinsic nsic ma math them emaatical cal va valu lues es su such ch as accuracy , confidence and systematic reasoning d)Stimulate learning environment that enhances effective learning inquiry-base and team work e)Develop mathematical knowledge in a way which increase students’ interest and confidence.
Introduction of integration In mathematics,integration is a technique of finding a function g ( x x ) the derivative of which, Dg ( x x ), ), is equal to a given function f ( x x ). ). This is indicated by the integral sign “∫,” as in ∫f ( x x ), ), usually called the indefinite integral of the function. (The symbol s ymbol dx is usually added, which merely identifies x as the variable.) The definite integral, integral , written with a and b called the limits of integration, is equal to g (b) − g (a), whereDg ( x x ) = f ( x ).Some antiderivatives can be calculated by merely recalling which function has a x ).Some given derivative, but the techniques of integration mostly involve classifying the functions according to which types of manipulations will change the function into a form the antiderivative of which can be more easily recognized. For example, if one is familiar with derivatives, the function 1/( x + 1) can be easily recognized as the derivative of loge( x x + 1). The anti-derivative of ( x 2 + x + 1)/( x x + 1) cannot be so easily recognized, but if written as x ( x x + 1)/( x x + 1) + 1/( x x + 1) = x + 1/( x x + 1), it then can be 2 recognized as the derivative of x x /2 + loge( x x + 1). One useful aid for integration is the theorem known as integration by parts. parts . In symbols, the rule is ∫ fDg = fg − ∫gDf. That is, if a function is the product of two other functions, f and one that can be recognized as the derivative of some function g , then the original problem can be solved if i f one can , then ∫
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Definition The process of finding a function, given its derivative, is called anti-differentiation (or integration). If F '('( x ), we say F ( x ). x ) = f ( x x ), x ) is an anti-derivative of f ( x x ). Examples F ( x x ) =cos x is an anti-derivative of sin x , and e x is an anti-derivative of e x . Note that if F ( x x ) is an anti-derivative of f ( x x ) then F ( x x ) + c , where c is a constant (called the constant of integration) is also an anti-derivative of F ( x ), as the derivative of a x ), constant function is 0. In fact they are the only anti-derivatives of F ( x x ). ).
We write f ( x x ) dx = F ( x x ) + c . if F '('( x x ) = f ( x x ) . We call this the indefinite integral of f ( x x ) . Thus in order to find the indefinite integral of a function, you need to be familiar with the techniques of differentiation of differentiation..
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HISTORY Over 2000 years ago, Archimedes (287-212 BC) found formulas for the surface areas and volumes of solids such as the sphere, the cone, and the paraboloid. His method of integration was remarkably modern considering that he did not have algebra, the function concept, or even the decimal representation of numbers. Leibniz (1646-1716) and Newton (1642-1727) independently independently discovered calculus. Their key idea was that differentiation and integration undo each other. Using this symbolic connection, they were able to solve an enormous number of important problems in mathematics, physics, and astronomy. Fourier (1768-1830) studied heat conduction with a series of trigonometric terms to represent functions. Fourier series and integral transforms have applications today in fields as far apart as medicine, linguistics, and music. Gauss (1777-1855) made the first table of integrals, and with many others continued to apply integrals in the mathematical and physical sciences. Cauchy (1789-1857) (1789-1857) took integrals to the complex domain. Riemann (1826-1866) and Lebesgue (18751941) put definite integration on a firm logical foundation. Liouville (1809-1882) created a framework for constructive integration by finding out when indefinite integrals of elementary functions are again elementary functions. Hermite (1822-1901) found an algorithm for integrating rational functions. In the 1940s Ostrowski extended this algorithm to rational expressions involving the logarithm. In the 20th century before computers, mathematicians developed the theory of integration and applied it to write tables of integrals i ntegrals and integral transforms. Among Among these mathematicians were Watson, Titchmarsh, Barnes, Mellin, Meijer, Grobner, Hofreiter, Erdelyi, Lewin, Luke, Magnus, Apelblat, Oberhettinger, Gradshteyn, Ryzhik, Exton, Srivastava, Prudnikov, Brychkov, and Marichev. In 1969 Risch made the major breakthrough in algorithmic indefinite integration when he published his work on the general theory and practice of integrating elementary elementary functions. His algorithm does not automatically apply to all classes of elementary functions because at the heart of it there is a hard differential equation that needs needs to be solved. Efforts since then have been directed at handling this equation algorithmically for various sets of elementary functions. These efforts efforts have led to an increasingly complete algorithmization algorithmization of the Risch scheme. In the 1980s some
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The capability for definite integration gained substantial substantial power in Mathematica, first released in 1988. Comprehensiveness and accuracy have been given strong consideration in the development of Mathematica and have been successfully accomplished in its integration code. Besides being able to replicate most of the results from well-known collections of integrals (and to find scores of mistakes and typographical typographical errors in them), Mathematica makes it possible to calculate countless new integrals not included in any published handbook.
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OBJECTIVE
Apply mathematics to everyday situation and appreciate the importance and the beauty of mathematics in everyday lives
To improve problem-solving skills, thinking skills, reasoning and mathematical communication.
To develop positive attitude and personalities and intrinsic mathematical values such as accuracy, confidence and systematic reasoning.
To stimulate learning environment that enhances effective effective learning, inquiry-based and our team work.
To develop mathematical knowledge in a way which increase our interest and confidence.
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PART 1
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Every time you graph an equation on a Cartesian coordinate system, you are using the work of René Descartes. Descartes, a French mathematician and philosopher, was born in La Haye, France (now named in his honor) on March 31, 1596. His parents taught him at home until he was 8 years old, when he entered the Jesuit college of La Flèche. There he continued his studies until he graduated at age 18. Descartes was an outstanding student at La Flèche, especially in mathematics. Because of his delicate health, his teachers allowed him to stay in bed until late morning. Despite missing most of his morning classes, Descartes was able to keep up with his studies. He would continue the habit of staying late in bed for his entire adult life. After graduating from La Flèche, Descartes traveled to Paris and eventually enrolled at the University of Poitiers. He graduated with a law degree in 1616 and then enlisted in a military school. In 1619, he joined the Bavarian army and spent the next nine years as a soldier, touring throughout much of Europe in between military campaigns. Descartes eventually settled in Holland, where he spent most of the rest of his life. There Descartes gave up a military career and decided d ecided on a life of mathematics and philosophy p hilosophy.. Descartes attempted to provide a philosophical foundation for the new mechanistic physics that was developing from the work of Copernicus and Galileo. He divided all things into two categories—mind and matter—and developed a dualistic philosophical system in which, although mind is subject to the will and does not follow physical laws, all matter must obey the same mechanistic laws. The philosophical system that Descartes developed, known as Cartesian philosophy, philosophy, was based on skepticism and asserted that all reliable knowledge must be built up by the use of reason through logical analysis. Cartesian philosophy was influential in the ultimate success of the Scientific Revolution and provides p rovides the foundation upon which most subsequent philosophical thought is grounded. Descartes published various treatises about philosophy and mathematics. In 1637 Descartes published his masterwork, Discourse on the Method of Reasoning Well and Seeking Truth in the Sciences. In Discourse, Di scourse, Descartes sought to explain everything in terms of matter and motion. Discourse contained three appendices, one on optics, one o ne on meteorology, and one titled La Géometrie (The Geometry). In La Géometrie, Descartes described what is now known as the system of Cartesian Coordinates, or coordinate geometry. In Descartes's system of coordinates, geometry and algebra were united for the first time to create what is known as analytic geometry. The Cartesian Coordinate System Cartesian coordinates are used to locate a point in space by giving its i ts relative distance from perpendicular intersecting lines. In coordinate geometry, geometry, all points, lines, and figures are drawn in a coordinate plane . By
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likewise be described and duplicated with coordinates. A straight line may also be represented on a coordinate grid. In the case of a straight line, every point on the line has coordinate values that must
The Cartesian coordinate system unites geometry and algebra, and is a universal system for unambiguous location of points. Applications range from computer animation to global positioning systems. satisfy a specific equation. The line in (c) may be expressed as y = 2x. The coordinates of every point on the line will satisfy the equation y = 2x, as for example, point A (1, 2) and point B (2, 4). More complex equations are used to represent circles, ellipses, and curved lines. Other Contributions La Géometrie made Descartes famous throughout Europe. He continued to publish his philosophy, philosophy, detailing how to acquire accurate knowledge. His philosophy is sometimes summed up in his statement, "I think, therefore I am." Descartes also made a number of other contributions to mathematics. He discovered the Law of Angular Deficiency for all polyhedrons and was the first to offer a quantifiable explanation of rainbows. In La Géometrie, Descartes introduced a familiar mathematics symbol, a raised number to indicate an exponent . The expression 4 × 4 × 4 × 4 × 4 may be written as 4 5 using Descartes's notation. He also instituted using x, y, and z for unknowns in an equation. In 1649, Descartes accepted an invitation from Queen Christina to travel to Sweden to be the royal tutor. Unfortunately for Descartes, the queen expected to be tutored while she did her exercises at 5:00 A.M. in an unheated library. library. Descartes had been used to a lifetime of sleeping late, and the new routine was much too rigorous for him. After only a few weeks of this regimen, Descartes contracted pneumonia and died on February
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PART
2
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Method 1 a)1.Divide area p into 3 segments.2 triangles and 1 rectangle. Triangle 1:
=1/2 x 2 x1 = 1m2
Triangle 2:
=1/2 x 4 x3 = 6m2
Rectangle 1: =2 x 4
= 8m2 P= 16 8 m2 P= 15m2 2.Divide area Q into 2 segments.1 triangle and 1 rectangle. Triangle 1:
=1/2 x 4 x 3 = 6m2
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Method 2 Coordinate geometry method:
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Area Q
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Method 3 Integration method Area P:
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=
dx ∫ 4 −2x 10dx ∫ 7 2dx ∫ 30 4x 3 3 4
=
[ ] 2x 3
2
3 [ − x 2 10 ] 4 [ 22 ] 7 0 3 4
= (6-0)+(24-21)+(14-8) = 6+3+6 =
15m 2
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Area Q
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Area R
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Verification answer by Geo Gebra : Area P
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Area Q
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Area R
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b)The mathematics society wishes to fence up the remaining sides of the region P. Determine the length of the fence required: from the diagram:
• 2
AE
=6 −2 23 2
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c) It is impossible for the society to carry out the fencing with an allocation of RM250.00. This is because: RM250.00+ RM25/m =10m
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Triangle 2
Calculate the length. Using Pythagoras theorem. L=
4.22− 6
2
2
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