Calculus 2, Part 1 Of 2: Integrals With Applications

Integral calculus in one variable: theory and applications for computing area between curves, curve length, and volumes

What you'll learn

How to solve problems concerning integrals of real-valued functions of 1 variable (illustrated with 419 solved problems) and why these methods work.

The concept of antiderivative / primitive function / indefinite integral of a function, and computing such integrals in a process reverse to differentiation.

Integration by parts as the Product Rule in reverse with many examples of its applications.

Integration by substitution as the Chain Rule in reverse with many examples of its applications.

Integration of rational functions with help of partial fraction decomposition.

Various types of trigonometric integrals and how to handle them.

Direct and inverse substitutions; various types of trigonometric substitutions.

The tangent half-angle substitution (universal trigonometric substitution).

Euler's substitutions.

Triangle substitutions.

Riemann integral (definite integral): its definition and geometrical interpretation in terms of area.

An example of a function that is not Riemann integrable (the characteristic function of the set Q, restricted to [0,1]).

Oscillatory sums; Cauchy criterion of (Riemann) integrability.

Sequential characterisation of (Riemann) integrability.

Proof of uniform continuity of continuous functions on a closed bounded interval.

Integrability of continuous functions on closed intervals.

Integration by inspection: Riemann integrals of odd (or: even) functions over compact and symmetric-to-zero intervals.

Integration by inspection: evaluating some definite integrals with help of areas known from geometry.

Fundamental Theorem of Calculus (FTC) in two parts, with a proof.

Applications of Fundamental Theorem of Calculus in Calc 2 and Calc3.

Application of FTC for computing derivatives of functions defined with help of Riemann integrals with variable (one or both) limits of integration.

Application of FTC for computing limits of sequences that can be interpreted as Riemann sums for some integrable functions.

The Mean-Value Theorem for integrals with proof and with a geometrical interpretation; the concept of a mean value of a function on an interval.

Applications of Riemann integrals: (signed) area between graphs of functions and the x-axis, area between curves defined by two continuous functions.

Applications of Riemann integrals: rotational volume.

Applications of Riemann integrals: rotational area.

Applications of Riemann integrals: curve length.

Improper integrals of the first kind (integration over an unbounded interval).

Improper integrals of the second kind (integration of unbounded functions).

Comparison criteria for determining whether an improper integral is convergent or not.

Requirements

Precalculus (Basic notions, Polynomials and rational functions, Trigonometry, Exponentials and logarithms)

Calculus 1: Limits and continuity (or equivalent)

Calculus 1: Derivatives with applications (or equivalent)

You are always welcome with your questions. If something in the lectures is unclear, please, ask. It is best to use QA, so that all the other students can see my additional explanations about the unclear topics. Remember: you are never alone with your doubts, and it is to everybody's advantage if you ask your questions on the forum.

Description

Calculus 2, part 1 of 2: Integrals with applicationsSingle variable calculusS1. Introduction to the courseYou will learn: about the content of this course and about importance of Integral Calculus. The purpose of this section is not to teach you all the details (this comes later in the course) but to show you the big picture.S2. Basic formulas for differentiation in reverseYou will learn: the concept of antiderivative (primitive function, indefinite integral); formulas for the derivatives of basic elementary functions in reverse.S3. Integration by parts: Product Rule in reverseYou will learn: understand and apply the technique of integration called "integration by parts"; some very typical and intuitively clear examples (sine or cosine times a polynomial, the exponential function times a polynomial), less obvious examples (sine or cosine times the exponential function), mind-blowing examples (arctangent and logarithm), and other examples.S4. Change of variables: Chain Rule in reverseYou will learn: how to perform variable substitution in integrals and how to recognise that one should do just this.S5. Integrating rational functions: partial fraction decompositionYou will learn: how to integrate rational functions using partial fraction decomposition.S6. Trigonometric integralsYou will learn: how to compute integrals containing trigonometric functions with various methods, like for example using trigonometric identities, using the universal substitution (tangent of a half angle) or other substitutions that reduce our original problem to the computing of an integral of a rational function.S7. Direct and inverse substitution, and more integration techniquesYou will learn: Euler substitutions; the difference between direct and inverse substitution; triangle substitutions (trigonometric substitutions); some alternative methods (by undetermined coefficients) in cases where we earlier used integration by parts or variable substitution.S8. Problem solvingYou will learn: you will get an opportunity to practice the integration techniques you have learnt until now; you will also get a very brief introduction to initial value problems (topic that will be continued in a future ODE course, Ordinary Differential Equations).S9. Riemann integrals: definition and propertiesYou will learn: how to define Riemann integrals (definite integrals) and how they relate to the concept of area; partitions, Riemann (lower and upper) sums; integrable functions; properties of Riemann integrals; a proof of uniform continuity of continuous functions on a closed bounded interval; a proof of integrability of continuous functions (and of functions with a finite number of discontinuity points); monotonic functions; a famous example of a function that is not integrable; a formulation, proof and illustration of The Mean Value Theorem for integrals; mean value of a function over an interval.S10. Integration by inspectionYou will learn: how to determine the value of the integrals of some functions that describe known geometrical objects (discs, rectangles, triangles); properties of integrals of even and odd functions over intervals that are symmetric about the origin;  integrals of periodic functions.S11. Fundamental Theorem of CalculusYou will learn: formulation, proof and interpretation of The Fundamental Theorem of Calculus; how to use the theorem for: 1. evaluating Riemann integrals, 2. computing limits of sequences that can be interpreted as Riemann sums of some integrable functions, 3. computing derivatives of functions defined with help of integrals; some words about applications of The Fundamental Theorem of Calculus in Calculus 3 (Multivariable Calculus).S12. Area between curvesYou will learn: compute the area between two curves (graphs of continuous functions), in particular between graphs of continuous functions and the x-axis.S13. Arc lengthYou will learn: compute the arc length of pieces of the graph of differentiable functions.S14. Rotational volumeYou will learn: compute various types of volumes with different methods.S15. Surface areaYou will learn: compute the area of surfaces obtained after rotation of pieces of the graph of differentiable functions.S16. Improper integrals of the first kindYou will learn: evaluate integrals over infinite intervals.S17. Improper integrals of the second kindYou will learn: evaluate integrals over intervals that are not closed, where the integrand can be unbounded at (one or both of) the endpoints.S18. Comparison criteriaYou will learn: using comparison criteria for determining convergence of improper integrals by comparing them to some well-known improper integrals.S19. ExtrasYou will learn: about all the courses we offer. You will also get a glimpse into our plans for future courses, with approximate (very hypothetical!) release dates.Make sure that you check with your professor what parts of the course you will need for your final exam. Such things vary from country to country, from university to university, and they can even vary from year to year at the same university.A detailed description of the content of the course, with all the 261 videos and their titles, and with the texts of all the 419 problems solved during this course, is presented in the resource file “001 List_of_all_Videos_and_Problems_Calculus_2_p1.pdf” under video 1 ("Introduction to the course"). This content is also presented in video 1.

Overview

Section 1: Introduction

Lecture 1 Introduction to the course

Lecture 2 Two types of integrals, two ways to go

Lecture 3 My choices versus the choices in the book

Lecture 4 The spoiler you need to follow both paths

Lecture 5 Main integration techniques and where to find them

Lecture 6 Plenty of applications

Section 2: Basic formulas for differentiation in reverse

Lecture 7 Reverting differentiation in simple cases

Lecture 8 Some important facts about primitive functions

Lecture 9 Integrals of hyperbolic functions and some related stuff

Lecture 10 Linearity of integration

Lecture 11 Linearity of integration, Exercise 1

Lecture 12 Linearity of integration, Exercise 2

Lecture 13 Linearity of integration, Exercise 3

Lecture 14 Linearity of integration, Exercise 4

Lecture 15 Linearity of integration, Exercise 5

Lecture 16 Linearity of integration, Exercise 6

Lecture 17 Linearity of integration, Exercise 7

Lecture 18 Linearity of integration, Exercise 8

Lecture 19 Linearity of integration, Exercise 9

Lecture 20 Linearity of integration, Exercise 10

Lecture 21 Linearity of integration, Exercise 11

Lecture 22 Linearity of integration, Exercise 12

Lecture 23 Linearity of integration, Exercise 13

Lecture 24 Linearity of integration, Exercise 14

Lecture 25 Linearity of integration, Exercise 15

Lecture 26 A soft introduction to variable substitution

Lecture 27 Easy variable substitution, Exercise 16

Lecture 28 Easy variable substitution, Exercise 17

Lecture 29 Easy variable substitution, Exercise 18

Lecture 30 Easy variable substitution and some uneasy trigonometry, Problem 1

Lecture 31 Easy variable substitution and some uneasy trigonometry, Problem 2

Lecture 32 Easy variable substitution, Exercise 19

Lecture 33 Easy variable substitution, Exercise 20

Lecture 34 Logarithmic derivative and its charm

Lecture 35 Logarithmic derivative, Exercise 21

Lecture 36 Logarithmic derivative: three difficult and important examples, Problem 3

Lecture 37 The last one, Exercise 22

Section 3: Integration by parts: Product Rule in reverse

Lecture 38 Integration by parts: how it works and when to use it

Lecture 39 Integration by parts: Example 1

Lecture 40 Integration by parts: Example 2

Lecture 41 Integration by parts: Example 3

Lecture 42 Integration by parts: Example 4

Lecture 43 Integration by parts: Example 5

Lecture 44 Integration by parts: Example 6

Lecture 45 Integration by parts: Example 7

Lecture 46 What happens when the degree of the polynomial is higher

Lecture 47 Integration by parts, Exercise 1

Lecture 48 Integration by parts, Exercise 2

Lecture 49 Integration by parts, Exercise 3

Lecture 50 Integration by parts, Exercise 4

Lecture 51 Integration by parts, Exercise 5

Lecture 52 Integration by parts, Exercise 6

Section 4: Change of variables: Chain Rule in reverse

Lecture 53 Integration by substitution: how it works and when to use it, Example 1

Lecture 54 Easy substitutions from Section 2, Example 2

Lecture 55 Recognising (almost) derivatives, Example 3

Lecture 56 Recognising (almost) derivatives, Example 4

Lecture 57 Recognising (almost) derivatives, Example 5

Lecture 58 Recognising (almost) derivatives, Example 6

Lecture 59 Recognising (almost) derivatives, Example 7

Lecture 60 Recognising (almost) derivatives, Example 8

Lecture 61 Recognising (almost) derivatives, Example 9

Lecture 62 Recognising (almost) derivatives, Example 10

Lecture 63 Recognising (almost) derivatives, Example 11

Lecture 64 Recognising (almost) derivatives, Example 12

Lecture 65 Recognising (almost) derivatives, Example 13

Lecture 66 Back to the integral from V49 for n=-1, Example 14

Lecture 67 Back to the integral from V30, Problem 1

Lecture 68 Different results can happen: how to handle them, Problem 2

Lecture 69 Optional: Back to the integral of cosecant from V36, Problem 3

Lecture 70 A less obvious case, Problem 4

Lecture 71 A less obvious case, Problem 5

Lecture 72 Three examples related to the arctangent, Problem 6

Lecture 73 Back to arcsine, Problem 7

Lecture 74 A strange one, Problem 8

Lecture 75 Three examples with the square root of x

Section 5: Integrating rational functions: partial fraction decomposition

Lecture 76 Five key concepts needed for the integration of rational functions

Lecture 77 Prerequisites from Precalculus 2

Lecture 78 We have already worked with integrals of rational functions

Lecture 79 Integrals leading to the logarithm or to power functions, Formula 1

Lecture 80 Integrals leading to the logarithm or to power functions, Formula 2

Lecture 81 Integrals leading to the arctangent or to power functions, Formula 3

Lecture 82 Formula 3, Example 3

Lecture 83 Variable substitution and arctangent, Formula 4

Lecture 84 Variable substitution and arctangent, Formula 5

Lecture 85 Variable substitution, logarithm, and arctangent, Formula 6

Lecture 86 Method of Strategic Substitution in partial fraction decomposition

Lecture 87 Integration of rational functions, Exercise 1

Lecture 88 Integration of rational functions, Exercise 2

Lecture 89 Integration of rational functions, Exercise 3

Lecture 90 Integration of rational functions, Exercise 4

Lecture 91 Integration of rational functions, Exercise 5

Lecture 92 Integration of rational functions, Problem 1

Lecture 93 Integration of rational functions, Problem 2

Lecture 94 Integration of rational functions, Problem 3

Lecture 95 Integration of rational functions, Problem 4

Section 6: Trigonometric integrals

Lecture 96 Trigonometric formulas and where to find them

Lecture 97 Trigonometric integrals we have seen until now

Lecture 98 Power reduction formulas for computing integrals of the sine (or cosine) squared

Lecture 99 Integral of the cube of the sine or of the cosine, by change of variables

Lecture 100 Integral of the fourth power of the sine with help of repeated power reduction

Lecture 101 Recursive formulas for integrals of powers of the sine or of the cosine

Lecture 102 Powers of the sine or of the cosine versus the functions of multiple arguments

Lecture 103 Cubes in two ways, Exercise 1

Lecture 104 Fourth powers in two ways, Exercise 2

Lecture 105 The product of powers of the sine and the cosine

Lecture 106 The product of powers of the sine and the cosine, Exercise 3

Lecture 107 The product of powers of the sine and the cosine, Exercise 4

Lecture 108 Integrals involving secants and tangents, Exercise 5

Lecture 109 Recursive formulas for integrals of powers of the tangent or of the secant

Lecture 110 Product to sum formulas are good for integrals

Lecture 111 Rational expressions in two variables

Lecture 112 World’s sneakiest substitution: the universal substitution

Lecture 113 The universal substitution, Exercise 6

Lecture 114 The universal substitution, Exercise 7

Lecture 115 The universal substitution, Exercise 8

Lecture 116 The universal substitution, Exercise 9

Lecture 117 Rational expressions odd w.r.t. the sine, Exercise 10

Lecture 118 Rational expressions odd w.r.t. the sine, Exercise 11

Lecture 119 Rational expressions odd w.r.t. the sine, Exercise 12

Lecture 120 Rational expressions odd w.r.t. the cosine, Exercise 13

Lecture 121 Rational expressions even w.r.t. both variables, Exercise 14

Lecture 122 Rational expressions even w.r.t. both variables, Exercise 15

Lecture 123 The last one, just for fun, Problem 1

Section 7: Direct and inverse substitution, and more integration techniques

Lecture 124 Nobody will teach you all the integration techniques, but…

Lecture 125 Euler substitutions, why three cases are enough

Lecture 126 Euler substitutions, why they work

Lecture 127 Euler's substitution 1, an explanation

Lecture 128 Euler's substitution 1, an example

Lecture 129 Euler's substitution 2, an explanation

Lecture 130 Euler's substitution 2, an example

Lecture 131 Euler's substitution 3, an explanation

Lecture 132 Euler's substitution 3, an example

Lecture 133 Optional: A geometrical interpretation of Euler's substitutions 2 and 3

Lecture 134 Euler substitutions, Problem 1

Lecture 135 Euler substitutions, Problem 2

Lecture 136 Euler substitutions, Problem 3

Lecture 137 Rational expressions of rational powers

Lecture 138 Rational expressions of rational powers, Problem 4

Lecture 139 Rational expressions of rational powers, Problem 5

Lecture 140 Rational expressions of rational powers, Problem 6

Lecture 141 Rational expressions of rational powers, Problem 7

Lecture 142 Rational expressions of rational powers, an atypical one, Problem 8

Lecture 143 Direct (u) versus inverse substitution

Lecture 144 Back to trigonometric substitutions from Section 6: reference triangles

Lecture 145 Three triangle substitutions

Lecture 146 Triangle substitutions, Case 1

Lecture 147 Triangle substitution 1, an example

Lecture 148 Triangle substitution 1, Problem 9

Lecture 149 Triangle substitution 1, Problem 10

Lecture 150 Triangle substitutions, Case 2

Lecture 151 Triangle substitution 2, an example

Lecture 152 Triangle substitution 2, Problem 11

Lecture 153 Triangle substitution 2, Problem 12

Lecture 154 Triangle substitutions, Case 3

Lecture 155 Triangle substitution 3, an example

Lecture 156 Triangle substitution 3, Problem 13

Lecture 157 Undetermined coefficients

Lecture 158 Some remarks about equality of certain functions

Lecture 159 Undetermined coefficients instead of integration by parts, an example

Lecture 160 Undetermined coefficients instead of integration by parts, another example

Lecture 161 Optional: Our most complicated method

Lecture 162 Optional: Our most complicated method, an example

Lecture 163 Optional: Our most complicated method, Problem 14

Section 8: Problem solving

Lecture 164 Practice, practice, practice

Lecture 165 Integrals, Problem 1

Lecture 166 Integrals, Problem 2

Lecture 167 Integrals, Problem 3

Lecture 168 Integrals, Problem 4

Lecture 169 Integrals, Problem 5

Lecture 170 Integrals, Problem 6

Lecture 171 Integrals, Problem 7

Lecture 172 Integrals, Problem 8

Lecture 173 Integrals, Problem 9

Lecture 174 Integrals, Problem 10

Lecture 175 Integrals, Problem 11

Lecture 176 Integrals, Problem 12

Lecture 177 Integrals, Problem 13

Lecture 178 Integrals, Problem 14

Lecture 179 Integrals, Problem 15

Lecture 180 Integrals, Problem 16

Lecture 181 Integrals, Problem 17

Lecture 182 A very brief introduction to Initial-Value Problems (IVP)

Lecture 183 IVP: verifying solutions, an example

Lecture 184 IVP: finding solutions, an example

Lecture 185 Position, velocity, acceleration

Lecture 186 Position, velocity, acceleration, Problem 18

Lecture 187 IVP: Falling under gravity, Problem 19

Lecture 188 Direct versus inverse problems

Section 9: Riemann integrals: definition and properties

Lecture 189 From Geometry to Calculus, one more time

Lecture 190 The concept of area

Lecture 191 Our early example

Lecture 192 Integrability

Lecture 193 Refinements of partitions, and relations between upper and lower Riemann sums

Lecture 194 Integrable functions, an example

Lecture 195 Finally, an example of a function that is not integrable

Lecture 196 More practical tests for integrability (Cauchy, sequential)

Lecture 197 Application of the new test to the example from V191

Lecture 198 Continuous functions on compact intervals are uniformly continuous

Lecture 199 Continuous functions on compact intervals are Riemann integrable

Lecture 200 Monotone functions on compact intervals are Riemann integrable

Lecture 201 Some properties of oscillations and oscillatory sums

Lecture 202 Some properties of Riemann integrals

Lecture 203 Monotonicity of integrals

Lecture 204 Additivity of integration w.r.t. the interval

Lecture 205 Integrability of piecewise continuous functions

Lecture 206 Mean Value Theorem for integrals 1

Lecture 207 Mean Value Theorem for integrals 2

Lecture 208 Mean value of a continuous function over a compact interval

Section 10: Integration by inspection

Lecture 209 Finding values of some integrals with help of geometry

Lecture 210 Integrals of odd functions over compact and symmetric-to-zero intervals

Lecture 211 Integrals of even functions over compact and symmetric-to-zero intervals

Lecture 212 Integrals of periodic functions

Lecture 213 Some nice properties of the mean value

Lecture 214 Optional: What about integrals equal to zero?

Section 11: Fundamental Theorem of Calculus

Lecture 215 The connection between finding areas and finding antiderivatives

Lecture 216 Algebraic and transcendental functions

Lecture 217 What came first: the logarithm or the exponential?

Lecture 218 Function of the upper limit of integration: its continuity

Lecture 219 Fundamental Theorem of Calculus, part 1

Lecture 220 Fundamental Theorem of Calculus, part 2 (the evaluation theorem)

Lecture 221 Area of a disc; another method for computing integrals from V147, V151, and V155

Lecture 222 Evaluating integrals, another example

Lecture 223 Important integrals for future applications (Fourier series)

Lecture 224 Computing average values of functions on intervals, an example

Lecture 225 Integration by parts for Riemann integrals, two ways to go

Lecture 226 Integration by substitution for Riemann integrals, two ways to go

Lecture 227 An illustration for integration by substitution

Lecture 228 Applications of properties of integrals, an exercise

Lecture 229 Limits of some type of sequences, Example 1

Lecture 230 Limits of some type of sequences, Example 2

Lecture 231 Limits of some type of sequences, Example 3

Lecture 232 Limits of some type of sequences, Example 4

Lecture 233 Limits of some type of sequences, Example 5

Lecture 234 Differentiating functions defined with help of integrals, Example 1

Lecture 235 Differentiating functions defined with help of integrals, Example 2

Lecture 236 Future: Fundamental Theorems in Multivariable Calculus

Section 12: Area between curves

Lecture 237 Area between the graph of a function and the x-axis

Lecture 238 Another (than in V36, V108, V114) method for finding antiderivative of secant

Lecture 239 Area between two graphs

Section 13: Arc length

Lecture 240 Arc length: derivation of the formula, some examples

Lecture 241 Arc length, Problem 1

Lecture 242 Arc length, Problem 2

Section 14: Rotational volume

Lecture 243 Rotational volume, different situations and different methods

Lecture 244 The disk method: derivation and an example

Lecture 245 We are finally able to confirm two well-known formulas for volume

Lecture 246 The washer method: derivation and an example

Lecture 247 Cylindrical shells: derivation and an example

Lecture 248 Cylindrical shells for a domain between two graphs

Section 15: Surface area

Lecture 249 Rotational surface area: derivation and two examples

Lecture 250 Rotational surface area, Problem 1

Lecture 251 Rotational surface area, Problem 2

Section 16: Improper integrals of the first kind

Lecture 252 Improper integrals, an introduction

Lecture 253 Improper integrals of the first kind, Problem 1

Lecture 254 Improper integrals of the first kind, Problem 2

Section 17: Improper integrals of the second kind

Lecture 255 Improper integrals of the second kind

Lecture 256 Improper integrals of the second kind, Problem 1

Lecture 257 Improper integrals of the second kind, Problem 2

Section 18: Comparison criteria

Lecture 258 Comparison criteria for improper integrals of non-negative functions

Lecture 259 Comparison criteria, Problem 1

Lecture 260 Comparison criteria, Problem 2

Lecture 261 Wrap-up and some words about "Calculus 2, part 2: Sequences and series"

Section 19: Extras

Lecture 262 Bonus Lecture

University and college students wanting to learn Single Variable Calculus (or Real Analysis),High school students curious about university mathematics; the course is intended for purchase by adults for these students