Mastering Calculus Ii: Techniques, Applications & Theory

A complete guide to Calculus II — from integrals and differential equations to polar coordinates, sequences, and series

What you'll learn

Apply the Fundamental Theorem of Calculus to compute definite and indefinite integrals.

Master u-substitution, integration by parts, trigonometric substitution, and partial fractions.

Calculate areas between curves, volumes of solids, and work done by variable forces.

Use numerical integration techniques like the Trapezoidal Rule and Simpson’s Rule.

Solve differential equations, including separable and linear types.

Analyze parametric and polar functions, including arc length and surface area.

Understand sequences and series, including convergence tests and power series.

Explore Taylor polynomials, complex numbers, and Fourier series.

Requirements

A solid understanding of Calculus I (limits, derivatives, basic integrals).

Familiarity with algebra, trigonometry, and basic functions.

Motivation to learn and explore mathematical concepts deeply.

Description

Calculus II is often considered one of the most challenging yet rewarding courses in the undergraduate mathematics curriculum. This course is your comprehensive guide to mastering Calculus II, covering everything from foundational integration techniques to advanced applications in physics, engineering, and probability.Whether you're a university student, a STEM professional, or a curious learner, this course is designed to demystify complex concepts and help you build a deep, intuitive understanding of integral calculus and its many applications.We begin with a refresher on the Fundamental Theorem of Calculus, ensuring you have a solid grasp of how differentiation and integration are connected. From there, we dive into powerful techniques of integration such as u-substitution, integration by parts, trigonometric substitution, and partial fraction decomposition—each explained with step-by-step examples and visual aids.But this course goes far beyond computation. You'll learn how to apply integration to solve real-world problems: calculating areas between curves, volumes of solids, work done by variable forces, and even hydrostatic pressure. You'll explore numerical methods like the Trapezoidal Rule and Simpson’s Rule, and understand how to estimate errors and optimize accuracy.As you progress, you'll tackle differential equations, parametric and polar coordinates, and arc length and surface area—all essential tools for modeling dynamic systems and analyzing geometric shapes.The course culminates in a deep dive into sequences and series, including convergence tests, power series, and Taylor polynomial approximations. You'll even explore complex numbers and Fourier series, opening the door to advanced topics in engineering, signal processing, and applied mathematics.Each topic is broken down into digestible lessons, supported by intuitive explanations, worked examples, and practice problems to reinforce your learning. Whether you're preparing for exams, refreshing your skills, or building a foundation for future study, this course will empower you to think like a mathematician and apply calculus with confidence.By the end of this course, you'll be able to:Tackle complex integration problems with confidence.Apply calculus to solve real-world physics and engineering problems.Understand the behavior of functions through series and approximations.Build a strong foundation for multivariable calculus and beyond.

Overview

Section 1: Review and foundations

Lecture 1 The Fundamental Theorem of Calculus - Part 2

Lecture 2 Computing Integrals using the Fundamental Theorem of Calculus

Lecture 3 Finding Areas using the Fundamental Theorem of Calculus

Lecture 4 The Limitations of the Fundamental Theorem of Calculus

Section 2: Techniques of Integration: u-Substitution

Lecture 5 What is u-Substitution?

Lecture 6 u-Substitution and Indefinite Integrals

Lecture 7 Examples of Finding Antiderivatives Using u-Substitution

Lecture 8 u-Substitution and Definite Integrals

Section 3: Definite Integrals & Area Between Curves

Lecture 9 The Definite Integral

Lecture 10 Area Between the Curves

Lecture 11 Area Between Curves when Boundaries Aren't Given

Lecture 12 Area Between Two Curves when the Curve Cross Each Other

Section 4: Volume by Integration (Disc, Washer, Shell, Cross-Sections)

Lecture 13 The Disc Method

Lecture 14 Symmetry and the Disc Method

Lecture 15 The Washer Method

Lecture 16 The Volume of a Cone

Lecture 17 The Washer Method with a Non-Standard Axis of Revolution

Lecture 18 Cross-Sectional Slicing

Lecture 19 Volume of a Circular Solid with Square Cross-Sections

Lecture 20 The Volume of a Pyramid by Cross-sectional Slicing

Lecture 21 The Volume of a Cylindrical Wedge with Triangular Cross-Sections

Lecture 22 The Shell Method

Lecture 23 Examples of the Shell Method

Lecture 24 The Shell Method with a Non-Standard Axis

Lecture 25 The Theorem of Pappus

Section 5: Work & Applications of Integration

Lecture 26 Work

Lecture 27 Work and Variable Force

Lecture 28 Hooke's Law and Integration

Lecture 29 Lifting a Rope and Integration

Lecture 30 Leaky Buckets and Integration

Lecture 31 Pumping Water and Integration

Section 6: Average & Mean Value Theorems

Lecture 32 Average Value of a Function

Lecture 33 Mean Value Theorem for Integrals

Section 7: Integration by Parts & Advanced Techniques

Lecture 34 Integration by Parts

Lecture 35 Examples of Integration by Parts

Lecture 36 Integration by Parts and Definite Integrals

Lecture 37 Integration by Cycles

Lecture 38 Integration by Hope

Lecture 39 Combining Substitution and Integration by Parts

Section 8: Trigonometric Integrals

Lecture 40 Trigonometric Integrals where Cosine or Sine have an Odd Power

Lecture 41 Trigonometric Integrals where Cosine or Sine have an Even Power

Lecture 42 Recommendations for Integrating Functions involving Cosines and Sines

Lecture 43 Trigonometric Integrals where Cosine or Sine have Different Periods

Lecture 44 Trigonometric Integrals involving an Even Power of Secants

Lecture 45 Trigonometric Integrals involving an Odd Power of Tangents

Section 9: Integration Concepts & Reduction Techniques

Lecture 46 Why is "+C" so Important to Indefinite Integrals?

Lecture 47 Reduction of Integrals involving Powers of Tangent

Lecture 48 The Antiderivative of Secant

Lecture 49 Reduction of Integrals involving Powers of Secant

Lecture 50 Antiderivatives of Cotangent and Cosecant

Section 10: Trigonometric Substitution

Lecture 51 An Introduction to Trigonometric Substitution

Lecture 52 Trigonometric Substitution using Sine

Lecture 53 Area of an Ellipse

Lecture 54 Trigonometric Substitution using Tangent

Lecture 55 You Don't Always Have to Use Trigonometric Substitution

Lecture 56 Trigonometric Substitution using Secant

Lecture 57 Another Example of Trigonometric Substitution

Lecture 58 Trigonometric Substitution involving a Coefficient

Lecture 59 Trigonometric Substitution and Completing the Square

Section 11: Partial Fraction Decomposition

Lecture 60 Partial Fraction Decomposition

Lecture 61 Long Division and Integration

Lecture 62 Partial Fraction Decomposition - Linear System Approach

Lecture 63 Partial Fraction Decomposition - Annihilation Approach

Lecture 64 Partial Fraction Decomposition - Repeated Linear Factors

Lecture 65 Partial Fraction Decomposition - Irreducible Quadratic Factor

Lecture 66 Partial Fraction Decomposition and Trigonometric Substitution

Lecture 67 Partial Fraction Decomposition - Repeated Quadratic Factors

Section 12: Algebraic & Rationalizing Substitutions

Lecture 68 Using Algebraic Identities to Simplify Integrals

Lecture 69 Rationalizing Substitution

Lecture 70 Rationalizing Substitution with Radical Exponent

Lecture 71 Another Example of Rationalizing Substitutions

Lecture 72 Rationalizing Substitution with an Exponential

Section 13: General Integration Tutorials

Lecture 73 Review of Numerical Integration

Lecture 74 Approximating Integrals using the Midpoint and Trapezoidal Rules

Lecture 75 Error Bounds of Midpoint and Trapezoidal Rules

Lecture 76 Determining the Minimal Number of Subdivisions for the Trapezoidal and Midpoint

Lecture 77 Approximating an Integral using the Midpoint Rule and Estimating Its Error

Lecture 78 Simpson's Rule

Lecture 79 Error Bound for Simpson's Rule

Lecture 80 Approximating an Integral using Simpson's Rule and Estimating Its Error

Section 14: Improper Integrals

Lecture 81 Improper Integrals

Lecture 82 Improper Integrals with Infinite Discontinuities

Lecture 83 The Comparison Test for Improper Integrals

Section 15: Arc Length & Surface Area

Lecture 84 Arc Length

Lecture 85 Arc Length of a Semicubal Parabola

Lecture 86 Arc Length of a Parabola

Lecture 87 Arc Length and Numerical Integration

Lecture 88 Area of a Surface of Revolution

Lecture 89 Surface Area of a Semicircular Ring

Lecture 90 Surface Area of a Parabolic Band

Lecture 91 Surface Area and Numerical Integration

Lecture 92 The Theorem of Pappus and Area of a Surface of Revolution

Section 16: Hydrostatics & Integration

Lecture 93 Hydrostatic Force

Lecture 94 Hydrostatic Force and Integration

Lecture 95 Hydrostatic Force against a Circular Plate

Lecture 96 Hydrostatic Force and the Theorem of Pappus

Section 17: Centroids

Lecture 97 Centroids

Lecture 98 Centroids and Integration

Lecture 99 Centroid of a Semicircular Disc

Lecture 100 The Centroid of a Sinusoidal Region

Lecture 101 The Centroid of a Region Bounded between Two Curves

Section 18: Probability & Continuous Random Variables

Lecture 102 Probability and Continuous Random Variables

Lecture 103 Probability and Improper Integrals

Lecture 104 Expected Values and Continuous Random Variables

Section 19: Differential Equations

Lecture 105 An Introduction to Differential Equations

Lecture 106 The Initial Value Problem

Lecture 107 Differential Equations and Implicit Anti-Differentiation

Lecture 108 Solving the Initial Value Problem

Lecture 109 Separable Differential Equations

Lecture 110 Separable Differential Equations and Implicit Anti-Differentiation

Lecture 111 Separable Differential Equations and Saline Solutions

Lecture 112 Differential Equations and the Law of Natural Growth

Lecture 113 Differential Equations and the Law of Inhibited Growth

Lecture 114 Differential Equations and Logistic Growth

Lecture 115 Linear Differential Equations

Lecture 116 Linear Differential Equations and Integrating Factors

Lecture 117 Linear Differential Equations and Saline Solutions

Lecture 118 Parametric Equations

Section 20: Parametric Equations & Applications

Lecture 119 Parametric Equations and Conic Sections

Lecture 120 The Cycloid

Lecture 121 Parametric Equations and Derivatives

Lecture 122 Tangent Lines of the Cycloid

Lecture 123 Parametric Functions and Area Under the Curve

Lecture 124 Parametric Functions and Surface Area

Section 21: Polar Coordinates & Applications

Lecture 125 Polar Coordinates

Lecture 126 Conversion Between Polar and Cartesian Coordinates

Lecture 127 Graphs of Polar Functions

Lecture 128 Graphs of Cardioids and Limaçons

Lecture 129 Polar Derivatives and Tangent Lines

Lecture 130 Polar Functions and Area Under the Curve

Lecture 131 Polar Functions and Area Between Two Curves

Lecture 132 Polar Functions and Arc Length

Section 22: Sequences

Lecture 133 Sequences

Lecture 134 Recursive Sequences

Lecture 135 General Forms of a Sequence

Lecture 136 Limits of Sequences

Lecture 137 Monotonic and Bounded Sequences

Lecture 138 The Monotone Convergence Theorem

Section 23: Series & Convergence Tests

Lecture 139 Series

Lecture 140 Geometric Sequences

Lecture 141 Geometric Sums

Lecture 142 Geometric Series

Lecture 143 Repeated Decimals and Geometric Series

Lecture 144 Geometric Series with an Unknown

Lecture 145 The Divergence Test

Lecture 146 Telescoping Series

Lecture 147 Properties of Series

Lecture 148 The Integral Test

Lecture 149 Using the Integral Test

Lecture 150 The p-Test

Lecture 151 Convergence of Series and Eventuality

Lecture 152 The Integral Test, Estimation, and Remainders

Lecture 153 The Comparison Test

Lecture 154 The Limit Comparison Test

Lecture 155 The Comparison Test, Estimation, and Remainders

Lecture 156 The Alternating Series Test

Lecture 157 Examples Using the Alternating Series Test

Lecture 158 The Alternating Series Test, Estimation, and Remainders

Lecture 159 Absolute and Conditional Convergence

Lecture 160 The Ratio Test

Lecture 161 The Root Test

Section 24: Infinite Series (Street Fighting Series)

Lecture 162 Strategies for Series

Lecture 163 Street Fighting Infinite Series - Part I

Lecture 164 Street Fighting Infinite Series - Part II

Lecture 165 Street Fighting Infinite Series - Part III

Lecture 166 Street Fighting Infinite Series - Part IV

Lecture 167 Street Fighting Infinite Series - Part V

Lecture 168 Street Fighting Infinite Series - Part VI

Lecture 169 Street Fighting Infinite Series - Part VII

Lecture 170 Street Fighting Infinite Series - Part VIII

Lecture 171 Street Fighting Infinite Series - Part IX

Section 25: Power Series

Lecture 172 Power Series

Lecture 173 Power Series and Intervals of Convergence

Lecture 174 Computing the Interval of Convergence of Power Series

Lecture 175 Derivatives and Integrals of Power Series

Lecture 176 Representing a Rational Function as a Power Series - Part I

Lecture 177 Representing a Rational Function as a Power Series - Part II

Lecture 178 Adjusting for the Numerator in a Power Series Representation

Section 26: Taylor Polynomial Approximations

Lecture 179 Approximating using Taylor Polynomials

Lecture 180 Approximating Exponentials using Taylor Polynomials

Lecture 181 Approximating Radicals using Taylor Polynomials

Section 27: Complex Numbers

Lecture 182 Complex Numbers

Lecture 183 Multiplication of Complex Numbers

Lecture 184 The Complex Plane

Lecture 185 Computing the Polar Form of a Complex Number

Section 28: Fourier Series

Lecture 186 Fourier Series

Lecture 187 The Fourier Series of the Square-wave Function

University students enrolled in Calculus II or equivalent courses.,STEM professionals needing a refresher in advanced calculus.,Anyone preparing for standardized tests (AP Calculus BC, GRE, etc.).,Curious learners interested in the mathematical foundations of physics, engineering, and data science.