Calculus Series . Math Thought Program

Calculus 2

Integral Calculus

This course continues your study of calculus by developing advanced integration techniques, sequences and series, parametric equations, and polar coordinates. The goal is not only to learn new methods, but to deepen your understanding of how integration works and why it matters. You will begin to see integration as a way of thinking about accumulation, area, and convergence with the same clarity you developed for limits and derivatives.

Calculus serves as a foundation across many fields. For that reason, the primary objective of this course, and the calculus sequence as a whole, is to strengthen your understanding of the core ideas first introduced in Calculus I. This course is designed to move you beyond computation and into genuine mathematical thinking. You will learn to connect ideas, interpret results, and approach problems with confidence rather than memorization.

By the end of the course, you will be proficient in a variety of analytical and numerical integration techniques. More importantly, you will understand how to use integration to model real situations, allowing mathematics to move from abstract procedures to meaningful application.

Prerequisite: Calculus 1 and AP Calculus AB

Self-Paced

Study Time: 12hr/wk

Master the art of integration.

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Video Lessons

102 Videos

📚

Topic Sections

12 Sections

🕑

Avg Per Lesson

~20 Minutes

📅

Duration

Self-paced

✍

Practice Sets

15+ Problems

🎓

Certificate

On Completion

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What You Need First

Calculus Series . Math Thought Program

Calculus 1

A brief review of inequalities, functions and plane analytic geometry; limits and continuity; the derivative and the differential; applications of dif...

Skills you will learn

Evaluate limits using algebraic, graphical, and numerical approaches
Apply the epsilon-delta definition to prove limit statements
Determine continuity of functions and classify discontinuities
Compute derivatives using the limit definition and differentiation rules
Apply the chain rule, product rule, and quotient rule fluently

Why it matters

Calculus 1 concepts are used directly in Calculus 2
Problem-solving techniques carry forward to more advanced topics
Differentiation rules are used constantly in integration techniques
Understanding limits is essential for convergence tests

~ Calculus 1

View course ›

AP Calculus . Math Thought Program

AP Calculus AB

A complete preparation course for the AP Calculus AB exam. Usually a 4/5 on the AP exam earns credit for Calculus 1, check with your institution for more information. Covers limits, derivatives, integrals, and the Fundamental Theorem of Calculus...

Skills you will learn

Evaluate limits analytically and interpret limit behavior graphically
Understand continuity and the Intermediate Value Theorem
Compute derivatives and apply differentiation rules
Solve AP-style related rates and optimization problems
Apply the Mean Value Theorem and analyze function behavior

Why it matters

Limits and continuity form the foundation for everything that follows in calculus
Derivatives allow you to describe change, motion, and behavior in real systems
Related rates and optimization connect mathematics to real-world decision making
The Mean Value Theorem explains why functions behave the way they do
These ideas prepare you to think, not just compute, on the AP exam and beyond

~ AP Calculus AB

View course ›

Course Topics & Lessons

[Cal2] (01): Integration Techniques

102 lessons

01.1

📖 Introduction

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01.2

📖 Introduction

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01.3

📖 Introduction

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01.4

📖 Introduction

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01.5

💬 Introduction

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01.6

📖 Introduction

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01.7

📖 Introduction

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01.8

📖 Introduction

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01.9

📖 Introduction

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01.10

📖 Introduction

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01.11

📖 Introduction

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01.12

📖 Introduction

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01.13

📖 Pythagorean Theorem Is The Key

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01.14

📖 Integration Technique Overview

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01.15

📖 Overview

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01.16

📖 IBP Overview

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01.17

📖 Trig Integrals Overview

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01.18

📖 Partial Fraction Overview

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01.19

📖 The Trapezoidal Rule

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01.20

📖 Improper Integral Overview

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01.21

📖 Simpson's Rule (3/8)

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01.22

📖 Error Propagation

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01.23

📖 Numerical Integration Overview

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01.24

📖 The Antiderivative

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01.25

🛠️ Riemann Sum APP

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01.26

📖 Left-Hand, Right-Hand, & Midpoint Rules

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01.27

📖 Basic Integration Properties

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01.28

📖 Thinking Through Problems

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01.29

Tabular Method

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01.30

Trigonometric Identities

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01.31

📖 Trig Sub Overview

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01.32

💬 Lecture: Linear Factors

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01.33

💬 Lecture: Type I - Unbounded

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01.34

Lesson 03: U-Sub [Part 01]

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01.35

💬 Lecture: Error Propagation

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01.36

📖 Why Should We Care About Integrals?

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01.37

📝 Practice Problems

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01.38

🛠️ Riemann Sum APP

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01.39

📖 Why Is Integration Difficult?

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01.40

Lesson 04: U-Sub [Part 02]

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01.41

💬 Lecture: Trig Integrals [Part 01]

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01.42

💬 Lecture: Sine Substitution

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01.43

💬 Lecture: Repeated Linear Factors

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01.44

📖 Definite Integrals

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01.45

💬 Lecture: Type II - Discontinuous

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01.46

💬 Lecture: The Trapezoidal Rule

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01.47

💬 Lecture: Simpson's Rule (1/3)

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01.48

Lecture: Basics of IBP

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01.49

📖 Fundamental Theorem of Calculus

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01.50

📹 Math In Motion: Example 01

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01.51

📹 Math In Motion: Example 01

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01.52

💬 Lecture: Trig Integrals [Part 02]

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01.53

💬 Lecture: Tangent Substitution

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01.54

💬 Lecture: Irreducible Quadratic Factors

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01.55

💬 Lecture: Improper Integral & Comparison Test

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01.56

💬 Lecture: Riemann Sums [Part 01]

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01.57

💬 Lecture: IBP [Part 02]

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01.58

Exciting Journey Ahead

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01.59

📖 Integrating Geometrically

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01.60

📝 Practice Problems

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01.61

📹 Math In Motion: Example 02

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01.62

💬 Lecture: Riemann Sums [Part 02]

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01.63

💬 Lecture: Converting Trigonometric Expressions

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01.64

📹 Math In Motion: Example 01

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01.65

📹 Math In Motion: Example 01

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01.66

p-Intergals Overview

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01.67

💬 Lecture: Secant Substitution

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01.68

📝 Practice Problems

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01.69

📖 Overview Of Techniques

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01.70

📝 Practice Problems

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01.71

📹 Math In Motion: Example 01

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01.72

📹 Math In Motion: Example 01

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01.73

📹 Math In Motion: Example 01

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01.74

Lesson 01: The Geometry of Integration

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01.75

📹 Math In Motion: Example 02

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01.76

📹 Math In Motion: Example 02

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01.77

📹 Math In Motion: Example 01

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01.78

🤔 CONCEPT CHECK

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01.79

Lesson 02: Definite Integral & FTC

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01.80

📹 Math In Motion: Example 03

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01.81

📹 Math In Motion: Example 03

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01.82

📹 Math In Motion: Example 02

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01.83

📹 Math In Motion: Example 02

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01.84

📹 Math In Motion: Example 02

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01.85

🃏 Flashcards: Numerical Formulas

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01.86

🃏 Summation Identities

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01.87

🤔 TEST YOUR MIGHT!!

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01.88

📝 Practice Problems

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01.89

📹 Math In Motion: Example 04

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01.90

📹 Math In Motion: Example 03

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01.91

🃏 Integration Identities

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01.92

📹 Math In Motion: Example 03

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01.93

📹 Math In Motion: Example 03

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01.94

📹 Math In Motion: Example 04

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01.95

📚 Practice Problems

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01.96

📝 Practice Problems

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01.97

🤔 CONCEPT CHECK

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01.98

📹 Math In Motion: Example 04

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01.99

🤔 Concept Check

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01.100

📹 Math In Motion: Example 05

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01.101

📝✏️ TEST YOUR MIGHT!!

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01.102

📝 Practice Problems

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01.103

📝 Practice Problems

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01.104

🤔 CONCEPT CHECK

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01.105

🤔 TEST YOUR MIGHT!!

🔒

01.106

🤔 TEST YOUR MIGHT!!

🔒

01.107

📹 Math In Motion: Example 06

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01.108

🤔 CONCEPT CHECK

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01.109

🤔 CONCEPT CHECK

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01.110

🤔 TEST YOUR MIGHT!!

🔒

01.111

📹 Math In Motion: Example 07

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01.112

🤔 TEST YOUR MIGHT!!

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01.113

🤔 TEST YOUR MIGHT!!

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01.114

📹 Math In Motion: Example 08

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01.115

📝 Practice Problems

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01.116

🤔 CONCEPT CHECK

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01.117

🤔 TEST YOUR MIGHT!!

🔒

[Cal2] (02) Integration Applications

29 lessons

02.1

📖 Introduction

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02.2

📖 Introduction

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02.3

📖 Introduction

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02.4

📖 Introduction

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02.5

📖 Introduction

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02.6

Introduction

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02.7

📖 Area Under A Curve

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02.8

📖 Nonrotational Volume

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02.9

📖 Pumping Liquids

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02.10

📖 Surface Area

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02.11

📖 Arc Length

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02.12

📖 Conclusion

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02.13

📖 Conclusion

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02.14

📖 Conclusion

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02.15

📖 Area Between Curves

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02.16

📖 Rotational Volume [Washers]

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02.17

📖 Conclusion

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02.18

💬 Lecture:

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02.19

💬 Lecture:

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02.20

💬 Lecture:

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02.21

📖 Rotational Volume [Shell]

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02.22

📖 Conclusion

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02.23

💬 Lecture:

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02.24

💬 Lecture:

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02.25

💬 Lecture:

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02.26

💬 Lecture:

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02.27

🤔 TEST YOUR MIGHT!!

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02.28

🤔 CONCEPT CHECK

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02.29

🤔 CONCEPT CHECK

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02.30

💬 Lecture:

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02.31

💬 Lecture:

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02.32

🤔 TEST YOUR MIGHT!!

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02.33

🤔 TEST YOUR MIGHT!!

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02.34

🤔 TEST YOUR MIGHT!!

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02.35

🤔 CONCEPT CHECK

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02.36

💬 Lecture:

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02.37

🤔 TEST YOUR MIGHT!!

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02.38

🤔 CONCEPT CHECK

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02.39

🤔 CONCEPT CHECK

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[Cal2] (03): Sequences

10 lessons

03.1

📖 Introduction

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03.2

📖 Arithmetic Sequences

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03.3

📖 Decrypting Patterns

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03.4

📖 Limits

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03.5

📖 Geometric Sequences

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03.6

📖 Monotonicity

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03.7

📖 Recursive Sequences

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03.8

📖 Bounded

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03.9

💬 Lecture: Sequence Introduction

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03.10

💬 Lecture: Sequence Limits and Monotonicity

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03.11

🤔 TEST YOUR MIGHT!!

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03.12

🤔 CONCEPT CHECK

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[Cal2] (04): Series Convergence Tests

32 lessons

04.1

📖 Introduction

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04.2

📖 Introduction

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04.3

📖 Introduction

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04.4

📖 Introduction

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04.5

📖 Introduction

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04.6

📖 Introduction

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04.7

⚙️ Convergence Flowchart

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04.8

📖 Divergence Test

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04.9

📖 Alternating Series Test

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04.10

📖 Telescopic Series

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04.11

📖 Direct Comparison Test [DCT]

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04.12

📖 Integral Test

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04.13

📖 Ratio Test

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04.14

🃏 Convergence Test [flashcards]

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04.15

📖 p-Series

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04.16

📖 Geometric Series

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04.17

📖 Limit Comparison Test [LCT]

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04.18

📖 Root Test

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04.19

📖 Error Terms

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04.20

💬 Lecture: Fundamentals

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04.21

🤔 TEST YOUR MIGHT!!

🔒

04.22

💬 Lecture: Integral Test

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04.23

💬 Lecture: DCT and LCT

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04.24

💬 Lecture: Ratio Test and Root Test

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04.25

📖 Rearrangements

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04.26

💬 Lecture: Divergence Test

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04.27

💬 Lecture: Telescopic Series

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04.28

🤔 CONCEPT CHECK

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04.29

💬 Lecture: Geometric Series

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04.30

🤔 CONCEPT CHECK

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04.31

🤔 CONCEPT CHECK

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04.32

🤔 CONCEPT CHECK

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04.33

🤔 CONCEPT CHECK

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04.34

🤔 CONCEPT CHECK

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04.35

Math Sketch: Geometric Series - Example 01

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04.36

🤔 TEST YOUR MIGHT!!

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04.37

🤔 TEST YOUR MIGHT!!

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04.38

🤔 TEST YOUR MIGHT!!

🔒

04.39

🤔 TEST YOUR MIGHT!!

🔒

04.40

🤔 TEST YOUR MIGHT!!

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04.41

💬 Lecture: Alternating Series

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04.42

Math Sketches: Example 01

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04.43

Math Sketch: Geometric Series - Example 02

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04.44

Math Sketches: Integral Test - Example 01

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04.45

🤔 CONCEPT CHECK

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04.46

🤔 TEST YOUR MIGHT!!

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[Cal2] (05): Power Series

24 lessons

05.1

Taylor Polynomial APP

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05.2

What Are Power Series?

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05.3

Lesson 43: Applications Sums with Taylor Polynomial

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05.4

Taylor Polynomials

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05.5

Euler's Identity

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05.6

Introduction

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05.7

Application: Computing Infinite Sums

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05.8

Flash Cards: Special Power Series

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05.9

Lesson 40: Power Series Fundamentals

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05.10

Geometric Series Manipulations

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05.11

Lesson 47: Discovering Euler’s Identity

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05.12

Lesson 42: Taylor Polynomials and Taylor Series

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05.13

Lesson 48: Power Series Applications and Using Euler’s Identity

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05.14

Interval and Radius of Convergence

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05.15

Lesson 41: Power Series & Geometric Representations

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05.16

Lesson 44: Powering Through Limits

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05.17

Taylor Series vs Maclaurin Series

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05.18

Application: Computing Limits

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05.19

Power Series: Natural Log

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05.20

Lecture 45: The Power Behind Solving Differential Equations

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05.21

Power Series: Arctangent

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05.22

Application: Solutions to Differential Equations

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05.23

Lesson 46: Powering Through Integrals

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05.24

Application: Computing Integrals

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[Cal2] (06): Parametric Equations

17 lessons

06.1

Parametric Equations Introduction

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06.2

Check List

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06.3

Polar Curves

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06.4

Polar Area

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06.5

Circles

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06.6

Graphing Parametric Equations

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06.7

Lesson 53: Fundamentals of Polar Coordinates

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06.8

Lesson 54: Graphing Polar Coordinates

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06.9

Limacon

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06.10

Lesson 51: Parametric Equation Fundamentals

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06.11

Lesson 55: Area of Polar Curves

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06.12

Rose Curves

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06.13

Lecture 56: Polar Tangent Lines

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06.14

Lesson 52: Calculus of Parametric Equation

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06.15

Lemniscate

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06.16

Solids of Revolution

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06.17

Special Curves

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Course Materials

● Required 2 books

Course Textbook

Thomas' Calculus: Early Transcendentals

Joel Hass, Christopher Heil, Przemyslaw Bogacki, Maurice D. Weir

ISBN: 9780137728626

Course Textbook

MyLab Math eText

18 Weeks Pearson ISBN: 9780137559794

How This Helps You

Mathematics

How Mathematics uses this course

Where you will use this directly

Abstract Algebra

You'll learn to recognize patterns in parametric equations and polar coordinates. This pattern recognition becomes crucial when working with group theory and ring structures in abstract algebra. You'll strengthen your ability to see connections between seemingly unrelated mathematical concepts.

Statistics & Probability

In Calculus 2, you'll master integration techniques that form the foundation for continuous probability distributions. These skills are essential for advanced statistical analysis, helping you model real-world phenomena with confidence.

Differential Equations

You'll develop powerful integration methods and series techniques that are crucial for solving differential equations. These mathematical tools help you model dynamic systems and predict how quantities change over time.

Logic & Problem Solving

Calculus 2 challenges you to work through multi-step integration problems and convergence proofs systematically. This develops your ability to break down complex challenges in any field into manageable steps.

Linear Algebra

You'll master series and sequences that provide essential background for understanding vector spaces and matrix operations. These analytical skills prepare you for the abstract thinking required in advanced mathematical concepts.

Instructor information will be available once you are enrolled.

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