Table of Contents

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Note on the Order of Sections

• Techniques of Integration
  1. Antiderivatives, Areas and the FTC
     1. Definition of Antiderivatives   1   2
        1. Definition of Antiderivatives
        2. Non-Uniqueness of Antiderivatives
     2. Antiderivative Rules   1   2   3   4
        1. Algebraic Operations
        2. Exp and Log Functions
        3. Trig Functions
        4. Hyperbolic Trig Functions (Optional)
     3. From Acceleration to Velocity and Position   1   2   3
        1. In General
        2. Motion with Constant Acceleration
        3. Ballistic Motion
     4. Differential Equations and Initial Value Problems   1
        1. Differential Equations and Initial Value Problems
     5. Area as an Antiderivative   1
        1. Area as an Antiderivative
     6. Fundamental Theorem of Calculus for Area   1
        1. Fundamental Theorem of Calculus for Area
     7. Exercises   (HW)   1
        1. Exercises
  2. Riemann Sums, Integrals and the FTC
     1. Definition of Riemann Sums   1   2   3
        1. Right Riemann Sums and Areas
        2. General Riemann Sums and Integrals (Optional)
        3. Limits of Riemann Sums (Optional)
     2. Fundamental Theorem of Calculus for Integrals   1   2
        1. The \(1^\text{st}\) and \(2^\text{nd}\) Fundamental Theorems
        2. Leibniz's Method
     3. Indefinite Integrals   1
        1. Indefinite Integrals
     4. Integration Rules and Properties   1   2   3
        1. Algebraic Operations
        2. Special Functions
        3. Definite Integral Properties
     5. Exercises   (HW)   1
        1. Exercises
  3. Integration by Substitution
     1. Indefinite Integrals   1   2
        1. Indefinite Integrals
        2. Adjusting the Differential
     2. Definite Integrals   1   2
        1. Definite Integrals - Substituting Back
        2. Definite Integrals - Changing Limits
     3. Exercises   (HW)   1
        1. Exercises
  4. Integration by Parts Available as Sample   (Problems include Area and Average Value)
     1. Indefinite Integrals   1   2
        1. Indefinite Integrals
        2. Integration by Parts with Multiple Steps
     2. Choosing the 'Parts'   1   2
        1. Choosing the 'Parts'
           1. Polynomial Examples
           2. Log Examples
           3. Inverse Trig Examples
        2. Integrals of Special Functions
     3. Definite Integrals   1
        1. Definite Integrals
     4. Recurrence of the Integral   1   2
        1. Solving for the Integral
        2. Reduction Formulas
     5. Exercises   (HW)   1
        1. Exercises
  5. Numerical Integration
     1. Approximation Techniques   1
        1. Approximation Techniques
     2. Left, Right, and Middle Riemann Sums   1   2   3   4
        1. Left, Right, and Middle Riemann Sums
        2. Example with a Known Antiderivative
        3. Example with an Unknown Antiderivative
        4. Example with Tabular Data
     3. Trapezoid Rule   1   2   3   4
        1. Trapezoid Rule
        2. Example with a Known Antiderivative
        3. Example with an Unknown Antiderivative
        4. Example with Tabular Data
     4. Simpson's Rule   1   2   3   4
        1. Simpson's Rule
           1. Area of the Best Fit Parabola - Proof
        2. Example with a Known Antiderivative
        3. Example with an Unknown Antiderivative
        4. Example with Tabular Data
     5. Error Bounds   1   2   3   4
        1. Mean Value Theorems
           1. First Degree Proof
           2. Higher Degree Proofs
        2. Left and Right Riemann Sums
           1. Left Riemann Sum Proof
           2. Right Riemann Sum Proof
        3. Trapezoid Rule and Middle Riemann Sum
           1. Examples to Bound the Error
           2. Examples to Guarantee a Bound
        4. Simpson's Rule
           1. Examples to Bound the Error
           2. Examples to Guarantee a Bound
     6. Exercises   (HW)   1
        1. Exercises
  6. Trigonometric Integrals
     1. Powers of Sine and Cosine   1   2   3
        1. One First Power
        2. One Odd Power - Easy Cases
        3. Both Even Powers - Hard Cases
     2. Powers of Tangent and Secant   1   2
        1. Easy Cases
        2. Hard Cases
     3. Converting Products to Sums   1
        1. Converting Products to Sums
     4. Exercises   (HW)   1
        1. Exercises
  7. Trigonometric Substitution Available as Sample   (Problems include Arc Length and Surface Area)
     1. Substitutions for \(x^2+1\) and \(x^2-1\)   1   2   3
        1. Substitutions for \(x^2+1\) - Tangent Substitutions
        2. Substitutions for \(1-x^2\) (or \(x^2-1\)) - Sine Substitutions
        3. Substitutions for \(x^2-1\) (or \(1-x^2\)) - Secant Substitutions
     2. Substitutions for \(a^2 x^2+b^2\) and \(a^2 x^2-b^2\)   1   2
        1. Substitutions for \(a^2 x^2+b^2\)
        2. Substitutions for \(a^2 x^2-b^2\)
     3. Completing Squares   1
        1. Completing Squares
     4. Hyperbolic Trig Substitutions (Optional)   1
        1. Hyperbolic Trig Substitutions (Optional)
     5. Exercises   (HW)   1
        1. Exercises
  8. Improper Integrals   (Problems include Area, Arc Length, Surface Area and Volume)
     1. Areas which Extend to Infinity   1
        1. Areas Which Extend to Infinity
     2. Integrals Improper at an Infinite Endpoint   1   2
        1. Definitions
        2. Comparison Test
     3. Integrals Improper at a Finite Endpoint   1   2
        1. Definitions
           1. Relation between \(\displaystyle\int\dfrac{1}{x^p}\,dx\) and \(\displaystyle\int\dfrac{1}{\sqrt[p]{x}}\,dx\)
        2. Comparison Test
     4. Other Improper Integrals   1   2
        1. Integrals Improper at an Interior Point
        2. Integrals Improper at Both Endpoints
     5. Exercises   (HW)   1
        1. Exercises
  9. Partial Fractions Available as Sample
     1. General Decompositions   1   2
        1. Introduction
        2. General Decompositions
           1. Long Division of Polynomials - Review
           2. Removing a Polynomial Part
           3. Non-Repeated Linear Factors
           4. Repeated Linear Factors
           5. Non-Repeated Quadratic Factors
           6. Partial Fraction Decomposition - Repeated Quadratic Factors
     2. Finding Coefficients   1   2
        1. Finding Coefficients - Linear
        2. Finding Coefficients - Quadratic
           1. Complex Methods (Optional)
     3. Computing Integrals   1   2
        1. Computing Integrals
           1. General Formulas
        2. Integral Examples
     4. Exercises   (HW)   1
        1. Exercises
• Applications of Integration
  10. Area and Average Value Available as Sample   (Requires Substitution)
      1. Area Below a Curve   1
         1. Area Below a Curve
      2. Area Between Two Curves   1   2
         1. Area Between Two Curves
         2. Area Between Two Curves - In Pieces
      3. Area as a y-Integral   1
         1. Area as a y-Integral
      4. Average Value and Mean Value Theorem   1   2   3
         1. Average Value of a Function
         2. Average Value and Area
         3. Mean Value Theorem for Integrals
      5. Tutorial on Average Value   1
         1. Average Value of a Function of 1 Variable
      6. Exercises   (HW)   1
         1. Exercises
  11. Mass and Center of Mass Available as Sample   (Requires Parts)
      1. Total Mass   1
         1. Mass: Bar with Non-Uniform Density
      2. Center of Mass   1   2
         1. Center of Mass: Discrete Masses
         2. Center of Mass: Bar with Non-Uniform Density
      3. Moment of Inertia   1
         1. Moment of Inertia - Bar with Non-Uniform Density
      4. Electric Charge   1
         1. Electric Charge - Bar with Non-Uniform Charge Density
      5. Exercises   (HW)   1
         1. Exercises
  12. Arc Length and Surface Area   (Requires Trig Integrals)
      1. Parametric Curves and Slopes   1   2   3
         1. Parametric Curves
         2. Slopes of Parametric Curves
         3. Horizontal and Vertical Tangents
      2. Arc Length   1   2   3   4
         1. Arc Length of a Parametric Curve
         2. Arc Length of the Graph of a Function
         3. Arc Length - 3-Dimensional Space
         4. Differential of Arc Length
      3. Surface Area of Revolution   1   2   3
         1. Area of a Surface of Revolution
            1. Surface Area of a Frustrum of a Cone
         2. Area of a Surface of Revolution of a Parametric Curve
         3. Area of a Surface of Revolution of a Graph
      4. Exercises   (HW)   1
         1. Exercises
  13. Volume Available as Sample   (Requires Trig Substitutions)
      1. Volume by Slicing   1
         1. Volume by Slicing
            1. Volume by Slicing - Example 1
      2. Solids of Revolution   1   2
         1. Solids of Revolution
         2. Volumes of Revolution: Disks, Washers or Cylinders?
      3. Volume using Thin Disks   1
         1. Volume using Thin Disks
            1. Volume using Thin Disks - Example 1
            2. Volume using Thin Disks - Example 2
      4. Volume using Thin Washers   1
         1. Volume using Thin Washers
            1. Volume using Thin Washers - Example
      5. Volume using Thin Cylinders   1
         1. Volume using Thin Cylinders
            1. Volume using Thin Cylinders - Example
      6. Strategy for Computing Volumes of Revolution   1
         1. Strategy for Computing Volumes of Revolution
      7. Tutorial   1
         1. Volume of Revolution
      8. Exercises   (HW)   1
         1. Exercises
  14. Work and Fluid Force   (Requires Trig Substitutions)
      1. Work   1   2
         1. Pointlike Object, Variable Force
            1. Hooke's Law for Springs
            2. Gravity Near the Surface of the Earth
            3. Newton's Law of Gravity
            4. Coulomb's Law of Electricity
         2. Extended Object, Variable Distance
            1. Lifting a Massive Rope
            2. Pumping Water out of a Tank
      2. Fluid Force   1
         1. Fluid Force
      3. Exercises   (HW)   1
         1. Exercises
  15. Polar Coordinates Available as Sample   (Requires Trig Substitutions)
      1. Polar Coordinate System   1   2   3
         1. Polar Coordinate System
         2. Converting Between Rectangular and Polar Coordinates
         3. Polar Equations
      2. Graphs of Polar Equations   1   2   3   4
         1. Graphs of Polar Equations
         2. Plotting Polar Equations
         3. Converting to Rectangular
         4. Intersections of Polar Graphs
      3. Slope of a Polar Graph   1
         1. Slope of a Polar Graph
      4. Area Enclosed by a Polar Graph   1   2
         1. Area Enclosed by a Polar Graph
         2. Area Between Polar Graphs
      5. Arc Length of a Polar Graph   1
         1. Arc Length of a Polar Graph
      6. Exercises   (HW)   1
         1. Exercises
• Differential Equations
  16. Differential Equations and Initial Value Problems
      1. What is a Differential Equation?   1   2
         1. Order and Degree
         2. Solutions of Differential Equations
      2. Initial Value Problems   1
         1. Initial Value Problems
      3. Separable Differential Equations   1   2   3   4
         1. The Technique
            1. Why Separation of Variables Works
         2. Examples
         3. Solution Families
         4. Logarithmic Examples
      4. Linear Differential Equations   1   2   3
         1. The Technique
            1. How the Integrating Factor is Chosen
         2. Examples
         3. Logarithmic Examples
      5. Slope Fields   1   2
         1. Definition
         2. Construction
      6. Exercises   (HW)   1
         1. Exercises
  17. Applications of Differential Equations Available as Sample
      1. Exponential Growth and Decay   1
         1. Exponential Growth and Decay
      2. Newtonian Heating and Cooling   1
         1. Newtonian Heating and Cooling
      3. Mixing Problems   1
         1. Mixing Problems
      4. A Robot Car   1
         1. A Robot Car
      5. Falling near the Earth   1   2
         1. Free Fall
         2. Falling with Air Resistance
      6. Logistic Growth   1
         1. Logistic Growth
      7. Exercises   (HW)   1
         1. Exercises
• Sequences & Series
  18. Sequences Available as Sample
      1. Definition of Sequences   1   2
         1. Definition
         2. Plots
      2. Limits of Sequences   1   2   3   4   5
         1. Limits - Definition
         2. Limits From Plots
         3. Special Limits
         4. Limit Laws
            1. Examples
         5. Limits Without Limit Laws
            1. Deriving Indeterminate Forms
            2. Indeterminate Forms \(\dfrac{0}{0}\) and \(\dfrac{\infty}{\infty}\)
            3. Indeterminate Form \(0\cdot\infty\)
            4. Indeterminate Form \(\infty-\infty\)
            5. Indeterminate Forms \(0^0\) and \(1^\infty\) and \(\infty^0\)
      3. Bounded and Monotonic Sequences   1   2   3
         1. Bounded Sequences
         2. Monotonic Sequences
         3. Bounded, Monotonic Sequence Theorem
      4. Recursively Defined Sequences   1   2
         1. Recursively Defined Sequences
         2. Limits of Recursively Defined Sequences
            1. Examples
      5. Precise Definitions and Proofs   1   2   3   4
         1. Precise Definiton of Limits
         2. Precise Special Limits
         3. Precise Limit Laws
         4. Precise Bounded Monotonic Sequence Theorem
      6. Exercises   (HW)   1
         1. Exercises
  19. Series
      1. Definition of Series   1   2
         1. Definition
         2. The Sum of a Series
      2. Geometric Series   1   2   3
         1. Finite
         2. Infinite
         3. Applications
      3. Telescoping Series   1
         1. Telescoping Series
            1. Standard Examples
            2. Using Partial Fractions
            3. Second and Higher Order (Optional)
      4. Convergence and Divergence Tests   1   2
         1. Convergence and Divergence Tests
            1. Fundamental Constants Defined by Series
         2. nth Term Divergence Test
      5. Exercises   (HW)   1
         1. Exercises
  20. Convergence of Positive Series Available as Sample
      1. Overview   1
         1. Overview
      2. The Integral Test and \(p\)-Series   1   2
         1. The Integral Test
            1. Proof
         2. The \(p\)-Series
      3. The Simple Comparison Test   1
         1. The Simple Comparison Test
      4. The Limit Comparison Test   1   2
         1. The Limit Comparison Test
            1. Proof
         2. The Limit Comparison Test - Extreme Cases
      5. The Ratio Test   1
         1. The Ratio Test
            1. Proof
      6. The Root Test   1
         1. The Root Test
            1. Proof
      7. Approximating the Series   1   2   3   4
         1. Integral Bounds
         2. Improved Integral Bounds
         3. Comparison Bounds
         4. Improved Comparison Bounds
      8. Exercises   (HW)   1
         1. Exercises
  21. Convergence of General Series
      1. Overview   1
         1. Overview
      2. The Alternating Series Test   1
         1. The Alternating Series Test
            1. Justification and Proof
      3. The Absolute Convergence Test   1
         1. The Absolute Convergence Test
            1. Proof
      4. The Ratio Test   1
         1. The Ratio Test
            1. Proof
      5. The Root Test   1
         1. The Root Test
            1. Proof
      6. Approximating the Series   1   2
         1. Alternating Series Bound
         2. Improved Alternating Series Bound
      7. Exercises   (HW)   1
         1. Exercises
  22. Power Series
      1. What is a Power Series?   1   2
         1. Sequences and Series of Functions
         2. Center, Terms and Coefficients
      2. Radius of Convergence   1   2
         1. Non-Zero, Finite Cases
         2. Extreme Cases
      3. Interval of Convergence   1
         1. Interval of Convergence
      4. Functions Defined by Power Series   1
         1. Functions Defined by Power Series
            1. Example: The Bessel Function
      5. Constructing New Power Series from Old Ones   1   2
         1. Operations on Power Series
            1. Substitution
            2. Addition and Subtraction
            3. Multiplication by Polynomials
            4. Differentiation
            5. Integration
         2. Summing Numeric Series
      6. Exercises   (HW)   1
         1. Exercises
  23. Taylor Series Available as Sample
      1. Taylor Series Definition   1   2
         1. Taylor Series Definition
         2. Computing Taylor Series
      2. Standard Maclaurin Series   1
         1. Standard Maclaurin Series
      3. Constructing New Taylor Series from Old Ones   1
         1. Constructing New Taylor Series from Old Ones
      4. Taylor Polynomial Approximation   1
         1. Taylor Polynomial Approximation
            1. \(\sin x\) about \(x=0\)
            2. \(\ln x\) about \(x=1\)
            3. \(\sqrt[\large 3]{x}\) about \(x=8\)
      5. Taylor Remainder   1   2   3
         1. Formula and Inequality
            1. Proof
         2. Approximation Error
         3. Convergence Proofs
      6. Exercises   (HW)   1
         1. Exercises
  24. Applications of Taylor Series Available as Sample
      1. Limits   1
         1. Limits
      2. Derivatives   1
         1. Derivatives
      3. Integrals   1   2
         1. Integrals
         2. Integrals (Requiring the Taylor Bound on the Remainder)
      4. Numeric Series and Fundamental Constants   1   2
         1. Numeric Series
         2. Fundamental Constants
      5. Differential Equations   1
         1. Differential Equations
      6. Exercises   (HW)   1
         1. Exercises
  25. Sample Exams
      1. Communications Quiz   1
         1. Communications Quiz
      2. MATH 172H Spring 2022   1
         1. MATH 172H Spring 2022
      3. MATH 172 Fall 2021   1
         1. MATH 172 Fall 2021
      4. MATH 172 Spring 2021   1
         1. MATH 172 Spring 2021
      5. MATH 172H Spring 2021   1
         1. MATH 172H Spring 2021
      6. MATH 172 Fall 2020   1
         1. MATH 172 Fall 2020
      7. MATH 172 Spring 2020   1
         1. MATH 172 Spring 2020
      8. MATH 172H Spring 2020   1
         1. MATH 172H Spring 2020
      9. MATH 172H Fall 2019   1
         1. MATH 172H Fall 2019
      10. MATH 172 Spring 2019   1
          1. MATH 172 Spring 2019
      11. MATH 172H Spring 2019   1
          1. MATH 172H Spring 2019
      12. MATH 172 Spring 2018   1
          1. MATH 172 Spring 2018
      13. MATH 152H Spring 2017   1
          1. MATH 152H Spring 2017
      14. MATH 152 Spring 2016   1
          1. MATH 152 Spring 2016
      15. MATH 152H Spring 2016   1
          1. MATH 152H Spring 2016
      16. MATH 152 Spring 2014   1
          1. MATH 152 Spring 2014
      17. MATH 152H Spring 2014   1
          1. MATH 152H Spring 2014
      18. MATH 152 Spring 2012   1
          1. MATH 152 Spring 2012
      19. MATH 152H Spring 2012   1
          1. MATH 152H Spring 2012
      20. MATH 152H Fall 2008   1
          1. MATH 152H Fall 2008
      21. MATH 152H Spring 2006   1
          1. MATH 152H Spring 2006
      22. MATH 152 Spring 2005   1
          1. MATH 152 Spring 2005
      23. MATH 152W Spring 2005   1
          1. MATH 152W Spring 2005
      24. MATH 152W Spring 2002   1
          1. MATH 152W Spring 2002
      25. MATH 152W Fall 2001   1
          1. MATH 152W Fall 2001
      26. MATH 152W Fall 2000   1
          1. MATH 152W Fall 2000
      27. MATH 152 Spring 2000   1
          1. MATH 152 Spring 2000
      28. MATH 172 Spring 1999   1
          1. MATH 172 Spring 1999
      29. MATH 152 Fall 1997   1
          1. MATH 152 Fall 1997
      30. Exercises   (HW)   1
          1. Exercises