Contents

Integral calculus 3

LECTURE NO. 19. PRIMITIVE AND INDEFINITE INTEGRAL 3

POINT 1. PRIMITIVE 3

POINT 2. INDEFINITE INTEGRAL AND ITS PROPERTIES 4

POINT 3. INTEGRATION BY SUBSTITUTION (CHANGE OF VARIABLE) 7

POINT 4. INTEGRATION BY PARTS 11

LECTURE NO.20. CLASSES OF INTEGRABLE FUNCTIONS 14

POINT 1. RATIONAL FUNCTIONS (RATIONAL FRACTIONS) 14

POINT 2. TRIGONOMETRIC FUNCTIONS 17

POINT 3. IRRATIONAL FUNCTIONS 22

INDEFINITE INTEGRAL: Basic Terminology 27

LECTURE NO. 21. DEFINITE INTEGRAL 32

POINT 1. PROBLEMS LEADING TO THE CONCEPT OFA DEFINITE INTEGRAL 32

POINT 2. DEFINITE INTEGRAL 34

POINT 3. PROPERTIES OF A DEFINITE INTEGRAL 36

POINT 4. DEFINITE INTEGRAL AS A FUNCTION OF ITS UPPER VARIABLE LIMIT 39

POINT 5. NEWTON-LEIBNIZ FORMULA 41

POINT 6. MAIN METHODS OF EVALUATION A DEFINITE INTEGRAL 42

LECTURE NO.22. APPLICATIONS OF DEFINITE INTEGRAL 46

POINT 1. PROBLEM – SOLVING SCHEMES. AREAS 46

POINT 3. VOLUMES 54

POINT 4. ECONOMIC APPLICATIONS 56

LECTURE NO. 23. DEFINITE INTEGRAL: ADDITIONAL QUESTIONS 58

POINT 1. APPROXIMATE INTEGRATION 58

Rectangular Formulas 58

Trapezium Formula 59

Simpson’s formula (parabolic formula) 60

POINT 2. IMPROPER INTEGRALS 63

Improper integrals of the first kind 63

Improper integrals of the second kind 66

Convergence tests 69

POINT 3. EULER Г- FUNCTION 71

DEFINITE INTEGRAL: Basic Terminology 73

LECTURE NO. 24. DOUBLE INTEGRAL 79

POINT 1. DOUBLE INTEGRAL 79

POINT 2. EVALUATION OF A DOUBLE INTEGRAL IN CARTESIAN COORDINATES 81

POINT 3. IMPROPER DOUBLE INTEGRALS. POISSON FORMULA 87

POINT 4. DOUBLE INTEGRAL IN POLAR COORDINATES 88

DOUBLE INTEGRAL: Basic Terminology 92

CONTENTS 95

Integral calculus (Інтеґральне числення): Методичний посібник по вивченню розділу курсу ”Математичний аналіз” для студентів ДонНту (англійською мовою)

УКЛАДАЧ: Косолапов Юрій Федорович, кандидат фізико-математич-них наук, професор

ФОРМАТ 60×84 . Умовних друкарських аркушів

83000, м. Донецьк, вул. Артема, 58, ДонНТУ