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SMA 200: Calculus II Lecture Notes

Institution: Kenyatta University

Course: Bachelor of Science

Content Category: lecture notes

Posted By: davidkariuki1010

Document Type: DOCX

Number of Pages: 129

Price: KES 200
 
    

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Summary

CONTENTS
CHAPTER 1: Integration of Functions Using Basic Formulae 1
1.1 Introduction
1.2 Objectives of the Lesson
1.3 Meaning of Integration
1.4 Constant of Integration
1.5 The Notation
1.6 Basic Formula for Integration of Functions
1.7 Addition and Subtraction Rule in Integration
1.8 Integration of Constant Times a Function Worked examples
Exercise Summary Further Reading

CHAPTER 2: Integration of Functions using the Extension of the Basic Formulae 8

2.1 Introduction
2.2 Objective of the Lesson
2.3 Definition of Integrand
2.4 The Extension of the Basic Formulae
2.5 Proof of the Extension Formulae Worked examples
Exercise Summary Further Reading


CHAPTER 3: Integration using Substitutions 16
3.1 Introduction
3.2 Objectives of the Lesson
3.3 Integration using Substitutions
3.4 Type I
3.5 Type II
3.6 Type III
3.7 Integrand containing u and u’ in other forms, (eu u1, sin u u1) Worked examples
Exercise Summary Further Reading


CHAPTER 4: Integration by Parts 26
4.1 Introduction
4.2 Objectives of the Lesson
4.3 Formulae for Integration by Parts
4.4 Formula for Integration of ln x Worked examples
Exercise Summary Further Reading

CHAPTER 5: Integration using Trigonometric Identities 34
5.1 Introduction
5.2 Objectives of the Lesson
5.3 Review of important Trigonometric identities
5.4 Integration using the three fundamental identities
5.5 Integration using the formulae for cos 2 x
5.6 Integration using the product formulae
5.7 Integration using sin 3 x and cos 3 x formulae
5.8 Integrand containing quadratic expressions in the Denominator. Worked examples
Exercise Summary

CHAPTER 6: Integration using Partial Fractions 47
6.1 Introduction
6.2 Objectives of the Lesson
6.3 Integrand Containing only non-repeated linear Factors in the Denominator (Proper Fraction)
6.4 Integrand Containing linear Factors (non-repeated) in the Denominator (Proper Fraction)
6.5 Integrand Containing Quadratic Factors.
6.6 Integrand is an (Improper Fraction) Worked examples
Exercise Summary Further Reading


CHAPTER 7: Definite Integrals and the Fundamental Theorem of the Integral Calculus. 53
7.1 Introduction
7.2 Objectives of the Lesson
7.3 Meaning of the Definite Integral
7.4 Important Properties of Definite Integral
7.5 Definition of Indefinite Integral

7.6 The Fundamental Theorem of the Integral Calculus
7.7 Evaluation of Definite Integrals Worked examples
Exercise Summary Further Reading
CHAPTER 8: Area Under a Curve 61
8.1 Introduction
8.2 Objectives of the Lesson
8.3 Formula for the Area under a Curve
8.4 Area Below x – axis
8.5 Combined Area under a Curve
8.6 Area Between Two Intersecting Curves. Worked examples
Exercise Summary Further Reading

CHAPTER 9: Improper Integrals 71
9.1 Introduction
9.2 Objectives of the Lesson
9.3 Definition of Improper Integrals
9.4 Convergence and Divergence of the Improper Integrals of the First Kind
9.5 Definition of Absolute and Conditional Convergence
9.6 Cauchy’s Inequality for convergence of an Integral
9.7 Absolute Convergence of an Integral Implies Convergence
9.8 Convergence and Divergence of the Improper Integrals of the Second Kind.
9.9 The Cauchy’s Principal value of Improper Integrals with Singularities. Worked examples
Exercise
Summary of the Chapter Further Reading


CHAPTER 10: Length of arc of a Curve 83
10.1 Introduction
10.2 Objectives of the Lesson
10.3 Length of the Arc in Cartesian Form
10.4 Length of the Arc in Parametric Form
10.5 Length of the Arc in Polar Coordinates Worked examples
Exercise Summary Further Reading


CHAPTER 11: Volume and Surface Area of Solids of Revolution 93
11.1 Introduction
11.2 Objectives of the Lesson
11.3 The Formula for the Volume of Solids of Revolution
11.4 The Formula for the Surface Area of Solids of Revolution Worked examples
Exercise
Summary of the Chapter Further Reading

CHAPTER 12: Numerical Integration (Trapezoidal and Simpsons Rules)
102
12.1 Introduction
12.2 Objectives of the Lesson
12.3 Area of a Trapezium
12.4 Trapezoidal Rule
12.5 Principle of Simpson’s Rule
12.6 Simpsons Rule Worked example Exercise
Summary of the chapter Further Reading
ANSWERS 115
INDEX


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