MA141 Chapters 4 and 5 versionf16 - NCSU Calculus Volume I John E Franke John R Griggs and Larry K Norris Calculus I Last update 2 c 2014-16 J E Franke

MA141 Chapters 4 and 5 versionf16 - NCSU Calculus Volume I...

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NCSU Calculus, Volume I John E. Franke, John R. Griggs and Larry K. Norris August 10, 2016
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Calculus I 2 c 2014-16 J. E. Franke, J. R. Griggs and L. K. Norris Last update: August 10, 2016
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Contents 4 Integration 5 4.1 Areas and Riemann Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.1.2 Answers to Selected Exercises . . . . . . . . . . . . . . . . . . . . . . 18 4.2 Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.2.2 Answers to Selected Exercises . . . . . . . . . . . . . . . . . . . . . . 31 4.3 Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . . . . . . . 32 4.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.3.2 Answers to Selected Exercises . . . . . . . . . . . . . . . . . . . . . . 40 4.4 Method of Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.4.2 Answers to Selected Exercises . . . . . . . . . . . . . . . . . . . . . . 50 4.5 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.5.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.5.2 Answers to Selected Exercises . . . . . . . . . . . . . . . . . . . . . . 61 5 Applications of Integration 63 5.1 Area Between Two Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.1.2 Answers to Selected Exercises . . . . . . . . . . . . . . . . . . . . . . 77 5.2 Volumes of Solids of Revolution . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.2.1 Disk Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.2.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.2.3 Answers to Selected Exercises . . . . . . . . . . . . . . . . . . . . . . 86 5.2.4 Washer Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.2.6 Answers to Selected Exercises . . . . . . . . . . . . . . . . . . . . . . 95 5.2.7 Cylindrical Shell Method . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.2.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.2.9 Answers to Selected Exercises . . . . . . . . . . . . . . . . . . . . . . 100 Index 101 Calculus I 3 c 2014-16 J. E. Franke, J. R. Griggs and L. K. Norris Last update: August 10, 2016
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CONTENTS Calculus I 4 c 2014-16 J. E. Franke, J. R. Griggs and L. K. Norris Last update: August 10, 2016
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Chapter 4 Integration Comments and suggestions are welcome! In this chapter we make a transition from indefinite integration to definite integration. To do this we take a side trip to the concept of the area under a curve . The purpose of this discussion is to motivate and connect the previously developed concepts in Chapter 3 to the new ones we develop in Section 4.2. The Fundamental Theorem of Calculus provides the connection between these concepts. The last sections are devoted to the first two of the many techniques of integration covered in calculus. 4.1 Areas and Riemann Sums The development of the definite integral involves quite a few sums. To compactly express these sums, we will use summation notation . If we have the sum of n numbers a 1 + a 2 + a 3 + · · · + a n the notation n X i =1 a i can be used to represent this sum succinctly. That is n X i =1 a i = a 1 + a 2 + a 3 + · · · + a n The Greek capital letter sigma Σ indicates a sum and the symbol a i represents the i th term. The variable i is called the index of summation . The values 1 and n indicate the beginning and ending values for i - taking on all consecutive integers between 1 and n inclusive. Letters other than i can be used and not all sums begin with i = 1. Example 1. Evaluate the sum 5 X i =1 4 i 2 Calculus I 5 c 2014-16 J. E. Franke, J. R. Griggs and L. K. Norris Last update: August 10, 2016
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4.1. AREAS AND RIEMANN SUMS Solution: 5 X i =1 4 i 2 = 4(1) 2 + 4(2) 2 + 4(3) 2 + 4(4) 2 + 4(5) 2 = 4 + 16 + 36 + 64 + 100 = 220 The nature of these sums allows for some algebraic simplifications. The following properties can be proved using the expanded form of each sum. Theorem 1. Algebra of Summation Let n be a natural number, a i and b i be real numbers for i from 1 to n , and c R , then 1 .
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