## Click to download the playlist as a pdf: Calculus II (Integral Calculus)

Approximating Area Under a Curve

Sigma Notation / Summation Notation

Ex: Area of a Parallelogram on the Coordinate Plane

Ex: Area of a Trapezoid on the Coordinate Plane

Area Under a Graph

Ex: Approximate Distance Traveled From a Table Using Area

Ex 1: Find the Area Under a Curve Using a Geometric Formula (Rectangle)

Ex 2: Find the Area Under a Curve Using a Geometric Formula (Triangle)

Ex 3: Find the Area Under a Curve Using a Geometric Formula (Trapezoid)

Determining Area Under Graphs Using Geometric Formulas

Ex: Evaluate a Definite Integral Using Area from a Graph

Ex: Definite Integration Using Geometric Formula (Line Above and Below X-Axis)

Ex: Definite Integration of an Absolute Value Function Using Geometric Formula

Ex: Evaluate a Definite Integral Using a Geometric Formula (Semicircle)

Ex: Accumulation of Area Under a Function Using Geometric Formulas

Ex: Application of Area Under a Function Using Geometric Formulas - Distance

Ex 1: Reimann Sum Using a Quadratic Function (Right Endpoints and Above x-axis)

Ex 2: Reimann Sum Using an Exponential Function (Left Endpoints and Above x-axis)

Ex 3: Reimann Sum Using a Quadratic Function (Right Endpoints and Above/Below x-axis)

Ex: Approximate the Area Under a Curve Using Rectangles (Left Using Graph)

Ex: Approximate the Area Under a Curve Using Rectangles (Right Using Graph)

Ex: Approximate the Area Under a Curve Using Rectangles (Midpoint Using Graph)

Area Under a Graph Using Rectangles - Application

Approximating Area Under a Graph Using Rectangles

Ex 1: Approximate the Area Under a Curve with 4 Left Sided Rectangles

Ex 2: Approximate the Area Under a Curve with 4 Right Sided Rectangles

Ex 3: Approximate the Area Under a Curve with 8 Left Sided Rectangles

Ex 4: Approximate the Area Under a Curve with 8 Right Sided Rectangles

The Antiderivative

Introduction to Antiderivatives and Indefinite Integration (No Trig)

The Antiderivative

Ex 1: Determine Antiderivatives

Ex 2: Determine Antiderivatives

Ex 3: Determine Antiderivatives

Ex 4: Determine Antiderivatives

Ex 5: Determine Antiderivatives

Ex 1: Antiderivative Concept - Given Information about f(x), Describe F(x)

Ex 2: Antiderivative Concept - Given Information about f(x), Describe F(x)

Ex: Find the Particular Solution to a Basic Differential Equation

Basic Antidifferentiation of Trigonometric Functions

Indefinite Integration

Ex: Basic Indefinite Integration (Polynomial, Exponential, Quotient)

Ex: Indefinite Integration with a Negative Exponent

Ex: Indefinite Integration Involving a Product

Ex: Indefinite Integration with a Variety of Terms

The Six Basic Trigonometric Integration Formulas

Indefinite Integration Using Basic Trig Integral Formulas: Part 1, Part 2

Integration Involving Inverse Trig Functions: Part 1, Part 2, Part 3

Definite Integral and The Fundamental Theorem of Calculus

The Definition of The Definite Integral

The Definite Integral

Ex: Setting Up a Definite Integral To Determine Area Under a Function

Ex: Definite Integral as Area Given a Graph (Function)

Ex: Definite Integral as Area Given a Graph (Function + Constant)

Ex: Definite Integral as Area Given a Graph (Constant*Function)

Ex: Evaluate Definite Integral Using Area Above and Below the x-axis

The Fundamental Theorem of Calculus

Proof of the Fundamental Theorem of Calculus (Part 2)

Ex: Evaluate a Definite Integral on the TI-84

Ex: Graph and Evaluate a Definite Integral on the TI84

Ex: Evaluate a Basic Definite Integral of a Constant Function Using the FTC

Ex: Evaluate a Basic Definite Integral of a Basic Linear Function Using the FTC

Ex: Evaluate a Basic Definite Integral of a Basic Quadratic Function Using the FTC

Ex: Evaluate a Basic Definite Integral of Cosine Using the FTC

Ex: Fundamental Theorem of Calculus Concept Check

Ex: Property of Definite Integral Subtraction

Ex: Property of Definite Integral Addition

Ex: Evaluate a Definite Integral of a Basic Quotient - Area Under a Curve

Ex: Evaluate a Definite Integral of a Polynomial

Local Maximum and Local Minimum of a Definite Integral Function (Accumulation Function)

Ex 1: Area Under a Constant Function Using Definite Integration

Ex 2: Area Under a Linear Function Using Definite Integration

Ex 3: Area Under a Quadratic Function Using Definite Integration

Ex 4: Area Under a Rational Function Using Definite Integration

Ex 5: Area Under a Piece Wise Defined Function Using Definite Integration

Ex: Definite Integral Involving a Basic Linear Function

Ex: Definite Integral Involving a Basic Rational Function

Ex: Definite Integral Involving a Rational Function Requiring Simplifying

Ex: Definite Integration Application - Cars Passing Through an Intersection

Ex: Definite Integration Involving a Basic Trig Function (nonnegative)

Ex: Definite Integration Involving a Basic Trig Function (above and below x-axis)

Determining the value of a definite integral on the graphing calculator

The Second Fundamental Theorem of Calculus

The Second Fundamental Theorem of Calculus

Proof of the Fundamental Theorem of Calculus (Part 1)

Ex 1: The Second Fundamental Theorem of Calculus

Ex 2: The Second Fundamental Theorem of Calculus (Reverse Order)

Ex 3: The Second Fundamental Theorem of Calculus

Ex 4: The Second Fundamental Theorem of Calculus with Chain Rule

Ex 5: The Second Fundamental Theorem of Calculus with Chain Rule

Ex 6: Second Fundamental Theorem of Calculus with Chain Rule

Ex 7: Second Fundamental Theorem of Calculus with Chain Rule

Ex: Evaluate a Definite Integral and the Derivative of an Integral Using a Graph

Applications of Definite Integration

Ex: Interpret the Meaning of Area Under a Function

Ex 1: Application of Definite Integration (Accumulated Sales)

Ex 2: Application of Definite Integration (Distance)

Ex: Definite Integration Application - Velocity and Distance

Ex 1: Integration Application - Work Lifting an Object

Ex 2: Integration Application - Work Lifting an Object and Cable

Ex: Find the Work Lifting a Leaking Bucket of Sand Given Mass

Ex: Find the Work Lifting a Leaking Bucket of Sand and Rope Given Mass

Ex: Find the Work Required to Stretch a Spring (Integration App)

Ex: Find the Force Required to Stretch a Spring (Integration App)

Ex: Definite Integral of Marginal Cost to find Total Cost

Properties of The Definite Integral

Properties of Definite Integrals and Average Value

The Mean Value Theorem for Integrals

Ex: Properties of Definite Integrals - Order of Integration

Ex: Properties of Definite Integrals - The Difference of Two Definite Integrals

Ex: Properties of Definite Integrals - Difference and Sum of Definite Integrals

Ex: Properties of Definite Integrals - Determine Limits of Integration

Ex: Properties of Definite Integrals - Zero Interval

Ex: Integration Application - Average Value of an Investment Account

Ex: Integration Application - Average Value of Temperature Function

Ex 1: Average Value of a Function

Ex 2: Average Value of a Trig Function

Ex: Integration Application - Average Value to Determine Average Coffee Temperature

Point of Equilibrium

Consumer and Producer Surplus

Present and Future Value: Part 1, Part 2

Area Bounded by Two Functions

Determining Area Between Two Curves - Integration Application

Area Between to Graphs

Ex 1: Find Area Between a Linear and Quadratic Function (respect to x)

Ex 2: Find Area Between a Linear and Exponential Function (respect to x)

Ex 3: Find Area Between Two Exponential Functions (respect to x)

Ex 4: Find Area Between Two Quadratic Functions (respect to x)

Ex 1: Area Bounded by Two Functions

Ex 2: Area Bounded by Two Functions (2 Regions)

Ex 3: Area Bounded by Two Trig Functions

Ex: Determine a Function Given The Area Between Two Functions

Integration by Substitution

Indefinite Integration Using Substitution

Integration by Substitution: Part 1, Part 2

Definite Integration Using Substitution

Ex 1: Indefinite Integration Using Substitution

Ex 2: Indefinite Integration Using Substitution

Ex 3: Indefinite Integration Using Substitution

Ex 4: Integration Using Substitution

Ex 5: Indefinite Integration Using Substitution

Ex 6: Indefinite Integration Using Substitution

Ex 7: Indefinite Integration Using Substitution

Ex 8: Indefinite Integration Using Substitution Involving Trig Functions

Ex 9: Indefinite Integration Using Substitution Involving Trig Functions

Ex: Evaluate a Indefinite Integral Using Substitution (Form e^u)

Ex: Evaluate a Indefinite Integral Using Substitution (Form ae^u with Decimals)

Indefinite Integration Using Substitution (Tough) Int(x^n*sqrt(x^(n-1)+c)

Ex 1: Definite Integration Using Substitution - Change Limits of Integration?

Ex 2: Definite Integration Using Substitution – Change Limits of Integration?

Ex: Evaluate a Definite Integral Using Substitution (Form e^u)

Ex: Evaluate a Definite Integral Using Substitution (Form ae^u with Decimals)

Ex: Evaluate a Definite Integral Using Substitution (Form 1/u)

Ex 1: Definite Integration Using Substitution

Ex 2: Definite Integration Using Substitution

Ex: Indefinite Integral Using Substitution Involving a Square Root

Ex: Indefinite Integral Using Substitution Involving a Rational Function I

Ex: Indefinite Integral Using Substitution Involving a Rational Function II

Definite Integration Using Substitution - Int(e^(1/x^n)/x^(n+1))

Ex: Indefinite Integral Using Substitution with Exponential and Sine

Ex: Definite Integration Using Substitution Involving Sine

Ex: Definite Integration Using Substitution Involving Exponential and Trig Functions

Ex: Indefinite Integral Involving Arcsine with Substitution

Indefinite integral: (sin(x))^2- Power Reducing Substitution

Indefinite Integral: (cos(2x))^2 - Power Reducing Substitution

Ex 1: Trigonometric Integration - Power Reducing Formula and U-Substitution

Ex 2: Trigonometric Integration - Power Reducing Formula and U-Substitution

Ex: Evaluate a Indefinite Integral Integration Tables and Substitution (cot^2(a^x))

Ex: Evaluate a Indefinite Integral Integration Tables and Substitution (sin^2(x^n))

Ex: Evaluate a Indefinite Integral Using Integration Tables

Integration Application: Area Using Parametric Equations – Ellipse

Integration Application: Area Using Parametric Equations - Cycloid

Integration by Parts

Integration by Parts: Basics

Ex: Integration by Parts - Basic Example

Integration by Parts

Integration by Parts: More Examples

Ex 1: Integration by Parts

Ex 2: Integration by Parts

Ex 3: Integration by Parts

Ex 4: Integration by Parts

Ex 5: Integration by Parts (Trig)

Ex 6: Integration by Parts Twice

Ex: Integration by Parts Involving a Radical and Natural Log

Ex: Integration by Parts Involving a Trig and Linear Function (x*cos(4x))

Ex: Integration by Parts - Definite Integral Involving a Quadratic and Natural Log Function

Ex: Definite Integral Using Integration by Parts in the Form x^n*ln(x)

Ex: Definite Integral Using Integration by Parts in the Form x^(n)*ln(bx)

Ex: Evaluate a Indefinite Integral Using Integration by Parts - Int(ln(ax+b),x)

Ex: Integration by Parts Twice Application

Ex: Integration by Parts Twice and Solving

Integration Involving Inverse

Trig Function and Integration Tables

Ex: Integration Tables - Basic Integration Involving sqrt(a^2-u^2)

Ex: Integration Tables - Basic Integration Involving a^2+u^2

Ex: Integration Tables - Basic Integration Involving a^2-u^2

Ex: Integration Tables - Integration Involving e^(ax)*sin(bx)

Ex: Integration Table - Integration Involving 1/u and a^2+u^2

Ex: Integration Tables - Integration Requiring U-Substitution sqrt(a^2-u^2)

Ex: Integration Tables - Integration Requiring U-substitution Involving sqrt(u^2+a^2)

Ex: Integration Tables - Integration Involving Requiring U-substitution Involving (tan(u))^n

Ex: Indefinite Integration Using U-Substitution Involving an Inverse Trig Function

Ex: Definite Integration Involving Inverse Tangent - 1/sqrt(a^2-u^2)

Ex: Definite Integration Involving Inverse Tangent - 1/(a^2+u^2)

Ex: Definite Integration Involving Inverse Tangent with U-Substitution - 1/(a^2+u^2)

Ex: Indefinite Integration Involving Arctangent Requiring U-sub and Completing the Square

Ex: Indefinite Integration Involving Arctangent Requiring Completing the Square

Numerical Integration

Ex 1: Numerical Integration - The Midpoint Rule

Ex 2: Numerical Integration - The Midpoint Rule (Fractions)

Trapezoidal Rule of Numerical Integration

Ex: Numerical Integration - The Trapezoid Rule

Trapezoid Rule Error - Numerical Integration Approximation

Trapezoid Rule - Determine n for a Given Accuracy

Simpson’s Rule of Numerical Integration

Ex: Simpson's Rule Using a Table of Values

Ex 1: Numerical Integration - Simpson's Rule

Ex 2: Estimate a Definite Integral Using Simpson's Rule (fractional subintervals)

Simpson's Rule Error - Numerical Integration Approximation

Simpson's Rule - Determine n for a Given Accuracy

Improper Integrals

Improper Integral

Ex 1: Improper Integrals

Ex 2: Improper Integrals

Ex 3: Improper Integrals

Ex 4: Improper Integrals and Area

Ex: Area Using Improper Integrals

Ex 1: Improper Integral - Infinite Interval (-inf,+inf)

Ex 2: Improper Integral - Infinite Interval (-inf, constant)

Ex 3: Improper Integral - Infinite Interval (-inf,+inf)

Ex 1: Improper Integral - Discontinuous Integrand

Ex 2: Improper Integral - Discontinuous Integrand

Ex: Improper Integral Involving Rational Function to Find Area Under a Curve

Ex: Improper Integral Involving Function with Rational Exponent to Find Area Under Curve

Introduction to Differential Equations

Introduction to Differential Equations

Ex: Determine Which Functions Are Solutions to a Differential Equation

Ex: Determine Which Function is a Solution to a Second Order Differential Equation

Ex: Verify a Solution to a Differential Equation and Find a Particular Solution

Ex: Find a Constant Function Solution to a Differential Equation

Ex: Find Two Exponential Function Solutions to a Differential Equation

Slope Fields

Ex: Determine Which Differential Equation Would Produce a Given Direction Field

Ex: Determine Direction Field Given a Solution to a Differential Equation

Ex: Select a Direction Field Given a Differential Equation Using Points

Differential Equations and Exponential Functions

Ex: Solve a Basic Initial Value Problem (Linear)

Ex: Solve a Basic Initial Value Problem (Exponential and Trig)

Solving a differential equation by separation of variables

Ex: Find the Particular Solution to a Basic Differential Equation

Ex 1: Initial Value Problem Using Separation of Variables (Square Root)

Ex 2: Initial Value Problem Using Separation of Variables (Square Root)

Ex 1: Initial Value Problem Using Separation of Variables Involving Natural Logarithm

Ex 2: Initial Value Problem Using Separation of Variables Involving Natural Logarithm

Ex 1: Initial Value Problem Using Separation of Variables in the form y' = ky

Ex 2: Initial Value Problem Using Separation of Variables in the Form y'=ay+b

Ex: Initial Value Problem in the Form y' = kx Using Shortcut

Ex: Initial Value Problem Using Separation of Variables in the Form y' = e^(ay+bx)

Ex: Solve an IVP Using Separation of Variables in the Form y'=(ax+b)/(xy^2)

Ex: Solve an IVP Using Separation of Variables in the Form y'=axy+bx

Applications of Integration: Business

Ex: Write a Differential Equation to Model the Change in a Bank Account

Ex: Limited Growth Differential Equation

Ex: Solve a Differential Equation that Models the Change in a Bank Account Balance

Ex: Logistic Growth Differential Equation

Ex: Complementary and Substitute Goods - Demand Function

Ex: Future Value of One Time Investment

Ex: Present Value of One Time Investment Given Future Value

Ex 1: Future Value of Continuous Money Flow

Ex 2: Continuous Money Flow needed for a Given Future Value

Ex: Present Value of Continuous Money Flow

Ex: Integration Application - Present Value for Business

Ex: Future and Present Value of Continuous Money Flow

Ex: Present Value of Perpetual Money Flow

Ex: Determine the Present Value of a Continuous Income Stream on the TI84 (Linear)

Ex: Point of Equilibrium

Consumer and Producer Surplus

Ex: Consumer Surplus (Linear)

Ex: Producer Surplus (Linear)

Ex: Consumer Surplus

Ex: Producer Surplus

Applications of Integration: Volume of Revolution

Determine Volume Of Solids by Slices - Integration Application

Ex 1: Volume of a Solid with Known Cross Section Using Integration - Volume by Slices

Ex 2: Volume of a Solid with Known Cross Section Using Integration - Volume by Slices

Ex 3: Volume of a Solid with Known Cross Section Using Integration - Volume by Slices

Ex 4: Volume of a Solid with Known Cross Section Using Integration - Volume by Slices

Ex 5: Volume of a Solid with Known Cross Section Using Integration - Pyramid

Ex: Volume of a Solid With Slices Parallel to X-axis (Triangle)

Volume of Revolution - The Disk Method

Ex: Volume of Revolution - Disk Method (y=x^(1/3)

Ex: Volume of Revolution - Disk Method (Quadratic Function)

Ex: Volume of Revolution - Disk Method (Exponential Function base e)

Ex 1: Volume of Revolution Using the Disk Method (Rational Function about y = 1)

Ex 2: Volume of Revolution Using the Disk Method (Sine Squared Function)

Ex 3: Volume of Revolution Using the Disk Method (Exponential Function)

Volume of Revolution - The Washer Method about the x-axis

Volume of Revolution - The Washer Method about the y-axis

Volume of Revolution - The Washer Method NOT about the x or y axis

Ex 1: Volume of Revolution Using Washer Method About y = 3

Ex 2: Volume of Revolution Using Washer Method About y = 3

Ex 1: Volume of Revolution Using Washer Method About Y-Axis

Ex 2: Volume of Revolution Using Washer Method About Y-Axis

Ex: Volume of Revolution Using Washer Method About x=5

Volume of Revolution - The Shell Method about the x-axis

Volume of Revolution - The Shell Method about the y-axis

Ex: Determine a Volume of Revolution Using the Shell (tube) Method (Quadratic About y-axis)

Ex: Determine a Volume of Revolution Using the Shell (tubes) Method (y-axis) - Calculator

Volume of Revolution - The Shell Method NOT about x or y axis

Ex: Volume of Revolution Using the Shell Method (Basic Quadratic about y axis)

Ex: Volume of Revolution Using the Shell Method (Quadratic about y axis)

Ex: Volume of Revolution Using the Shell Method (Sine about y axis)

Ex: Volume of Revolution Using the Shell Method (Exponential about y axis)

Ex: Volume of Revolution Using Shell Method with Horizontal Axis (Not X-Axis)

Ex: Volume of Revolution Using Shell Method with Vertical Axis (Not Y-Axis)

Volume of Revolution - Comparing the Washer and Shell Method

Applications of Integration: Arc Length, Surface Area, Work, Force, Center of Mass

Arc Length – Part 1

Arc Length – Part 2

Ex: Find the Arc Length of a Linear Function

Ex: Find the Arc Length of a Radical Function (Rational Exponent)

Ex: Find the Arc Length of a Quadratic Function

Surface Area of Revolution – Part 1

Surface Area of Revolution – Part 2

Ex: Surface Area of Revolution - Linear Function

Ex: Surface Area of Revolution - Sine Function

Ex: Find the Surface Area of Revolution of a Cubic Function About the x-axis

Ex: Find the Surface Area of Revolution of a Square Root Function About the x-axis

Ex: Determine the Work Required to Pump Water Out of a Circular Cylinder

Ex: Find the Surface Area of Revolution of a Quadratic Function About y-axis (Respect to x)

Ex: Find the Surface Area of Revolution of a Cube Root Function About y-axis (Respect to y)

Ex: Determine the Work Required to Pump Water Out of Trough (Isosceles Triangle)

Ex: Determine the Work Required to Pump Water Out of Trough (Quadratic Cross Section)

Ex: Find the Work Lifting a Leaking Bucket of Sand Given Mass

Ex: Find the Work Lifting a Leaking Bucket of Sand and Rope Given Mass

Ex: Determine the Center of Mass of Three Point Masses on the Coordinate Plane

Ex: Find the Centroid of a Region Consisting of Three Rectangles

Ex: Find the Centroid of a Triangular Region on the Coordinate Plane

Ex: Find the Centroid of a Bounded Region Involving Two Quadratic Functions

Ex: Find the Centroid of a Bounded Region Involving the Sine Function Using the TI84

Ex: Find the Hydrostatic Force on a Horizontal Plate (No Calculus)

Ex: Find the Hydrostatic Force on a Vertical Plate in the Shape of an Isosceles Triangle

Ex: Find the Hydrostatic Force on a Dam in the Shape of a Degree 4 Polynomial

Ex: Find the Hydrostatic Force on a Semicircle Window Submerged in Water

Integration Involving Powers of Trigonometric Functions

Trig Integrals Involving Powers of Sine and Cosine: Part 1, Part 2

Trig Integrals Involving Powers of Secant and Tangent: Part 1, Part2

Ex: Integral Using Substitution with an Odd Power of Cosine

Ex: Integral Using Substitution with an Odd Power of Sine

Ex: Integral Using Substitution with an Odd Power of Tangent

Ex: Integral Using Substitution with an Even Power of Secant

Wallis’s Formula to Integrate Powers of Sine and Cosine on [0, pi/2]

Integration Using Partial Fractions

Partial Fraction Decomposition: Part 1, Part 2

Ex: Partial Fraction Decomposition - Degree 2 / Degree 3

Integration Using Partial Fraction Decomposition: Part 1, Part 2

Ex 1: Integration Using Partial Fraction Decomposition

Ex 2: Integration Using Partial Fraction Decomposition and Long Division

Ex: Indefinite Integral Requiring Partial Fraction Decomposition

Integration Using Trigonometric Substitution

Integration Involving Trigonometric Substitution: Part 1, Part 2, Part 3, Part 4

Ex 1: Integration Using Trigonometric Substitution

Ex 2: Integration Using Trigonometric Substitution

Ex 3: Integration Using Trigonometric Substitution

Ex 4: Integration Using Trigonometric Substitution

Ex 5: Integration Using Trigonometric Substitution

Ex 6: Integration Using Trigonometric Substitution

Ex: Integration Using Trigonometric Substitution and Completing the Square

Ex 1: Definite Integration Using Trigonometric Substitution

Ex 2: Definite Integration Using Trigonometric Substitution

Ex: Indefinite Integral in the form x^n*sqrt(a^2+x^2) Using Trigonometric Substitition

Ex: Indefinite Integral in the form x^n*sqrt(a^2+x^2) Using U-Substitition

Ex: Indefinite Integral in the form x^n*sqrt(a^2 - x^2) Using Trigonometric Substitition

Ex: Indefinite Integral in the form x^n*sqrt(a^2 - x^2) Using U-Substitition

Ex: Indefinite Integration in the Form sqrt(a^2-x^(2n))/x^(n+1) Using Trigonometric Substiution

Wallis’s Formula to Integrate Powers of Sine and Cosine on [0, pi/2]

Infinite Series

Introduction to Sequences

Arithmetic Sequences

Geometric Sequences

Ex: Find the Formula for a Geometric Sequence Given Terms

Sequences on the TI84 Graphing Calculator

Limits of a Sequence

Ex: Limit of a Sequence Using L'Hopital's Rule (Divergent

Ex: Limit of a Sequence (cos(n)/2^n)

Ex: Limit of a Sequence Using L'Hopital's Rule Twice (Convergent)

Ex: Limit of a Sequence Using L'Hopital's Rule (Convergent)

Ex: Limit of a Sequence (Num Degree Greater)

Ex 1: Limit of a Sequence (Linear/Linear)

Ex 2: Limit of a Sequence (Quadratic/Quadratic)

Ex: Limit of a Sequence (Num Degree Less)

The Squeeze Theorem

Arithmetic Series

Geometric Series

Introduction to Infinite Series

Infinite Series: The Nth Term Divergent Test

Infinite Geometric Series

Sequences and Series on the TI84

Graph Partial Sums of an Infinite Series on the TI84

Telescoping Series

Ex 1: Telescoping Series (Convergent)

Ex 2: Telescoping Series (Divergent)

Ex 3: Telescoping Series with Partial Fractions

The Harmonic Series

The Integral Test

Infinite Series: The Integral Test

Ex: Infinite Series - Integral Test (Rational Function and Convergent)

Ex: Infinite Series - Integral Test (Rational Function and Divergent)

Ex: Infinite Series - Integral Test (Radical and Divergent)

Ex: Infinite Series - Integral Test (Exponential and Convergent)

Ex: Infinite Series - Integral Test (Convergent Involving Arctangent)

Ex: Infinite Series - Integral Test Requiring Integration by Parts (Convergent)

The p-series Test

Infinite Series: The p-Series Test

Ex 1: Infinite Series - P Series Test (Convergent) and Find a Partial Sum

Ex 2: Infinite Series - P Series Test (Divergent) and Find Partial Sum

The Direct Comparison Test

Infinite Series: The Direct Comparison Test

Ex: Infinite Series - Direct Comparison Test (Convergent)

Ex: Infinite Series - Direct Comparison Test (Divergent)

Ex: Infinite Series - Direct Comparison Test (Inconclusive)

The Limit Comparison Test

Ex: Infinite Series - Limit Comparison Test (Convergent)

Ex: Infinite Series - Limit Comparison Test (Geometric, Divergent)

Ex: Infinite Series - Limit Comparison Test (Radical, Convergent)

Ex: Infinite Series - Limit Comparison Test (Divergent)

Ex: Infinite Series - Limit Comparison Test (Radical, Divergent)

Infinite Series: The Limit Comparison Test (Divergent)

Infinite Series: The Limit Comparison and Direct Comparison Tests

Infinite Series: The Limit Comparison and Ratio Tests - Part 1

Infinite Series: The Limit Comparison and Ratio Tests - Part 2

The Root Test

Infinite Series: The Root Test I

Infinite Series: The Root Test II

The Ratio Test

Ex 1: Infinite Series - The Root Test (Convergent)

Ex 2: Infinite Series - The Root Test (Divergent)

Ex 3: Infinite Series - The Root Test (Divergent)

Ex 4: Infinite Series - The Root Test (Convergent)

Ex 5: Infinite Series - The Root Test (Divergent)

Infinite Series: The Ratio Test I

Infinite Series: The Ratio Test II

Ex 1: Infinite Series - The Ratio Test (Convergent)

Ex 2: Infinite Series - The Ratio Test (Divergent)

Ex 3: Infinite Series - The Ratio Test (Convergent)

Ex 4: Infinite Series - The Ratio Test (Convergent)

The Alternating Series Test

Ex: Find a Partial Sum of a Alternating Series (Method #1)

Ex: Find a Partial Sum of a Alternating Series (Method #2)

Conditionally and Absolutely Convergent Series

Ex 1: Determine if a Series Is Conditionally Convergent, Absolutely Convergent, or Divergent

Ex 2: Determine if a Series Is Conditionally Convergent, Absolutely Convergent, or Divergent

Ex 3: Determine if a Series Is Conditionally Convergent, Absolutely Convergent, or Divergent

Ex 4: Determine if a Series Is Conditionally Convergent, Absolutely Convergent, or Divergent

Infinite Series: The Alternating Series Test

Ex: Apply Alternating Series to Infinite Series - Divergent

Ex: Determine if an Infinite Alternating Series Converges or Diverges (Convergent)

Ex: Determine if an Infinite Alternating Series Converges or Diverges (Divergent)

Ex 1: Determine if an Series and an Alternating Series Converge or Diverge

Ex 2: Determine if an Series and an Alternating Series Converge or Diverge

Ex: Find the Error When Using a Partial Sum to Estimate an Infinite Sum (Alternating Series)

Ex: Number of Terms Needed in Partial Sum to Estimate an Infinite Sum with a Given Error

Taylor Polynomials

Taylor’s Theorem with Remainder

Power Series

Power Series: Part 1, Part 2

Representing a Function as a Geometric Power Series: Part 1, Part 2

Ex 1: Interval of Convergence for Power Series (Centered at 0)

Ex 2: Interval of Convergence for Power Series (Centered at 0)

Ex 3: Interval of Convergence for Power Series (Centered at 0)

Ex 4: Interval of Convergence for Power Series (Centered at 0)

Ex 5: Interval of Convergence for Power Series (Not Centered at 0)

Ex 6: Interval of Convergence for Power Series (Not Centered at 0)

Taylor and Maclaurin Series

Using Power Series Tables – Part 1, Part 2

Ex 1: Maclaurin Series and Polynomial of cos(2x) / Find Approximation Error

Ex: Find the Taylor Series of x^3

Ex: Find the Taylor Series of e^x

Ex: Find a Degree One and Degree Two Maclaurin Polynomial

Determine the Maclaurin Series and Polynomial for Function in the Form a*cos(bx^2)

Determine the Maclaurin Series and Polynomial for Function in the Form ax^2*e^(bx)

Determine the Maclaurin Series and Polynomial for Function in the Form ax^2*sin(bx)

Ex: Determine a Taylor Polynomial for a Square Root Function

Ex: Find a Maclaurin Polynomial and Error of an Approximation - ln(cos(x))

Ex: Find a Maclaurin Polynomial and the Interval for a Given Error - cos(x)

Ex: Find a Maclaurin Polynomial and the Interval for a Given Error - ln(1+x)

Ex: Use a Maclaurin Polynomial for sin(bx^n) to Approximate an Integral

Ex 1: Find a Power Series to Represent a Rational Function

Ex 2: Find a Power Series to Represent a Rational Function

Ex 3: Find a Power Series to Represent a Power Series

Ex: Find a Power Series to Represent a Power Series Using a Product

Ex: Find a Power Series to Represent a arctan(x) Using Integration

Ex: Find a Power Series to Represent a Rational Function Using Differentiation

Differentiating and Integrating Using Power Series

Ex: Determine a Simplified Power Series for a Function Involving e^(ax)

Ex 1: Determine the Sum of a Power Series

Ex 2: Determine the Sum of a Power Series

Estimate a Definite Integral using a Power Series (Rational Function)

Parametric Equations

Introduction to Parametric Equations

Graphing Parametric Equations in the TI84

Converting Parametric Equation to Rectangular Form

Ex 1: Write Parametric Equations as a Cartesian Equation

Ex 2: Write Parametric Equations as a Cartesian Equation

Ex 3: Write Parametric Equations as a Cartesian Equation

Ex 4: Write Parametric Equations as a Cartesian Equation

Ex: Parametric Equations Modeling a Path Around a Circle

Ex: Parametric Equations for an Ellipse in Cartesian Form

Ex: Find Parametric Equations For Ellipse Using Sine And Cosine From a Graph

Find the Parametric Equations for a Line Segment Given an Orientation

Determine Which Parametric Equations Given Would Give the Graph of the Entire Unit Circle

Determine Which Parametric Equations Would Give the Graph of an Entire Line

Ex 1: Find the Parametric Equations for a Lissajous Curve

Ex 2: Find the Parametric Equations for a Lissajous Curve

Ex 3: Find the Parametric Equations for a Lissajous Curve

Ex 4: Find the Parametric Equations for a Lissajous Curve

Ex: Point on a Spoke of a Rotating Wheel - Find the Radius

The Derivative of Parametric Equations

Ex 1: Derivatives of Parametric Equations and Applications

Ex 2: Derivatives of Parametric Equations and Applications (Trig)

Ex 1: Equation of a Tangent Line to a Curve Given by Parametric Equations

Ex 2: Equation of a Tangent Line to a Curve Given by Parametric Equations

Ex 3: Equation of a Tangent Line to a Curve Given by Parametric Equations

Determine the Points Where the Tangent Lines are Horizontal or Vertical Using Parametric Equations

Second Derivative of Parametric Equations: Part 1, Part 2

Ex: Determine the First and Second Derivative Given Parametric Equations

First and Second Derivative of Parametric Equations - Concavity

Arc Length in Parametric Form

Ex 1: Determine the Arc Length of a Curve Given by Parametric Equations

Ex 2: Determine the Arc Length of a Curve Given by Parametric Equations

Find the Length of a Loop of a Curve Given by Parametric Equations

Area Under Parametric Curves

Surface Area of Revolution in Parametric Form

Ex 1: Surface Area of Revolution in Parametric Form

Ex 2: Surface Area of Revolution in Parametric Form

Polar Coordinates and Equations

Polar Coordinates

Ex: Convert Cartesian Coordinates to Polar Coordinates

Animation: Rectangular and Polar Coordinates

Converting Polar Equations to Rectangular Equations

Ex: Find the Rectangular and Polar Equation of a Circle From a Graph

Ex: Find the Polar Equation of a Circle With Center at the Origin

Ex: Find the Polar Equation for a Horizontal Line

Ex: Find the Polar Equation for a Line

Ex: Write a Polar Equations of a Line as a Cartesian (Rectangular) Equations

Ex: Find the Polar Equation for a Parabola

Ex: Find the Rectangular Equation of a Circle from a Polar Equation

Ex: Convert a Polar Equation to a Rectangular Equation

Graphing Polar Equations

Graph Polar Equations I

Graph Polar Equations II

Animation: Graph Polar Equations

Ex: Determine the Type of Conic Section Given a Polar Equation

Graph Conic Sections in Polar Form: Part 1, Part 2, Part 3

Conics in Polar Form and Graphing a Parabola in Polar Form

Graphing an Ellipse in Polar Form

Graphing a Hyperbola in Polar Form

Ex: Find the Intercepts and Foci of a Ellipse Given a Polar Equation

Ex: Find the Intercepts and Foci of a Hyperbola Given a Polar Equation

Ex: Find the Intercepts and Focus of a Parabola Given a Polar Equation

Derivatives and Integrals with Polar Equations

Ex: Determine the Slope of a Tangent Line to a Polar Curve at a Given Angle

Ex: Determine Where a Polar Curve Has a Horizontal Tangent Line

Area using Polar Coordinates: Part 1, Part 2, Part 3

Ex: Find the Area Bounded by a Polar Curve Over a Given Interval (Spiral)

Ex: Find the Area of a Inner Loop of a Limacon (Area Bounded by Polar Curve)

Ex: Find the Area of Petal of a Rose (Area Bounded by Polar Curve)

Area between Polar Curves: Part 1, Part 2

Ex: Find the Area of a Region Bounded by a Polar Curve (r=Acos(n*theta))

Ex 1: Find the Area of a Region Bounded by Two Polar Curves

Ex 2: Find the Area of a Region Bounded by Two Polar Curves

The Slope of a Tangent Line to a Polar Curve

Horizontal and Vertical Tangent Lines to a Polar Curve

Arc Length of a Polar Curve

Ex 1: Arc Length of a Polar Curve

Ex 2: Arc Length of a Polar Curve

Ex: Find the Perimeter of a Region Bounded by Two Polar Curves

Surface Area of Revolution of a Polar Curve

Vectors in 2D

Introduction to Vectors

Vector Operations

Unit Vector

Ex: Find the Sum of Two Vectors From a Graph (2 Dimensions)

Ex: 2D Vector Scalar Multiplication

Ex: Find the Unit Vector Given the Graph of a Vector in 2D

Ex: Find the Difference of Two Vectors in Component Form

Ex: Find the Sum of Two Vectors Given in Linear Combination Form

Ex: Find the Difference of Two Vector Given in Linear Combination Form

Ex: Find the Difference of Scalar Multiples of Vectors in 2D

Ex: Dot Product of Vectors - 2D

Ex: Dot Product of Vectors From a Graph - 2D

Ex: Find a Component of a Vector So Two Vectors are Orthogonal (Dot Product)

Ex 1: Find a Vector in Component Form Given an Angle and the Magnitude (30)

Ex 2: Find a Vector in Component Form Given an Angle and the Magnitude (45)

Ex 3: Find a Vector in Component Form Given an Angle and the Magnitude (60)

Ex 4: Find a Vector in Component Form Given an Angle and the Magnitude (90)

Ex 5: Find a Vector in Component Form Given an Angle and the Magnitude (180)

Ex: Find the Difference of Scalar Multiples of Vectors in 2D

Ex: Find the Magnitude of The Difference of Two Vectors and The Difference of Two Magnitudes

Find the Component Form of a Vector from the Graph of a Vector

Ex: Find the Direction and Magnitude of a Vector in Component Form

Find the Component Form of a Vector Given Magnitude and Direction

Ex: Write a Vector as a Combination of Two Vectors

Ex: Find the Net Force of Three Vectors and the Opposite Force

Ex: Find the Coordinates of a Rotated Point Using Vectors

Applications of Vectors

Applications of Vectors

Determining the Angle Between Two Vectors

Proof of the formula for the Angle Between Two Vectors

Vector Projection

Proof of the Vector Projection Formula

Ex: Vector Projection in Two Dimensions

Ex: Find the Angle of Intersection of Two Curves Using Vectors

Ex: Direction and Speed of a Plane in the Wind Using Vectors

Ex: Vector App: Find an Airplane Direction In The Wind To Fly Due North

Vector App: Find the Direction of a Ball Thrown From a Car

Ex: Vector App - Find the Resultant Vector of a 5 Direction Walk

Ex: Vector App - Find the Resultant Vector of a 2 Direction Walk

Vector Applications: Force and Work

Vectors in Space

Plotting Points in 3D

The Equations of the Coordinate Planes in R3

Graphing a Plane Using Intercepts

Ex: Determine the Distance Between a Point and a Coordinate Plane in R3

Ex: Equation of a Sphere Given the Center and Radius

The Equation of a Sphere

Ex: Find the Difference of Scalar Multiples of Two Vectors in 3D (Linear Combination Form)

Vectors in Space

Parallel Vectors

Ex: Dot Product of Vectors - 3D

Ex: Find the Component of a Vector so Two Vectors are Orthogonal (3D)

Ex: Find the Angle Between Two Vectors in Three Dimensions

Ex: Vector Projection in Three Dimensions

Ex: Find the Component Form of a Vector in Space Given the Initial and Terminal Point

Ex: Find a Unit Vector in the Direction of a Given Vector in 3D

Ex: Find the Magnitude of a Vector in 3D

Ex: Find the Sum of Scalar Multiples of Two Vectors in 3D (Component Form

Ex: Find the Difference of Scalar Multiples of Two Vectors in 3D (Linear Combination Form)

Vector Cross Products

Ex: Find the Cross Product of Two Vectors

Ex: Find Two Unit Vectors Orthogonal to Two Given Vectors

Ex 1: Properties of Cross Products - Cross Product of a Sum and Difference

Ex 2: Properties of Cross Products - Cross Product of a Sum and Difference

Ex: Find the Area of a Triangle Using Vectors - 3D

Ex: Find the Distance Between Two Points In Space

An Application of Cross Products: Torque

The Triple Scalar Product: Volume of a Parallelepiped

Parametric Equations of Lines in 3D

Equations of Planes and Lines in Space

Ex: Determine the Equation of a Plane Given a Point and Normal Vector

The Equation of a Plane in 3D Using Vectors

Graphing a Plane in 3D

Ex: Determine the Point of Intersection of a Plane and a Line.

Find an Equation of a Plane Containing a Line and Orthogonal to a Given Plane

Find an Equation of a Plane Containing a Given Point and the Intersection of Two Planes

Determine the Linear Equation of the Intersection to Two Planes

Determining the Angle Between Two Planes

Determining the Distance Between a Plane and a Point

Determining the Distance Between a Line and a Point

Ex: Find the Distance Between Two Parallel Planes

Ex: Find the Equation of the Plane Containing a Given Line and a Point Using Vectors

Ex: Find the Equation of a Plane Given Three Points in the Plane Using Vectors

Ex: Find the Equation of a Plane Given an Orthogonal Line (Parametric) and a Point

Ex: Find the Equation of Plane Containing a Line and Orthogonal to a Given Plane

Ex: Find the Equation of a Plane Given a Point in the Plane and a Parallel Plane

Ex: Find the Parametric Equations of a Line in Space Given Two Points on the Line

Ex: Find the Point Where a Line in 3D Intersects the xz-plane

Ex 1: Find the Parametric Equations of the Line of Intersection of Two Planes Using Vectors

Ex 2: Find the Parametric Equations of the Line of Intersection of Two Planes Using Vectors

Ex: Find the Parametric Equations of a Line Perpendicular to a Plane Through a Point

Ex: Find Parametric Equations of a Line in Space Parallel to a Vector Containing a Given Point

Quadric Surfaces, Cylindrical Coordinates and Spherical Coordinates

Cylindrical Surfaces

Introduction to Quadric Surfaces

The Ellipsoid

The Hyperboloid of One Sheet

The Hyperboloid of Two Sheets

The Elliptical Cone

The Elliptical Paraboloid

The Hyperbolic Paraboloid

Surfaces of Revolution

Cylindrical Coordinates

Converting Between Cylindrical and Rectangular Equations

Spherical Coordinates

Converting Between Spherical and Rectangular Equations

Ex 1: Convert Cartesian Coordinates to Spherical Coordinates

Ex 2: Convert Cartesian Coordinates to Spherical Coordinates

Ex 1: Convert Spherical Coordinates to Cartesian Coordinates

Ex 2: Convert Spherical Coordinates to Cartesian Coordinates

Ex 1: Convert Cartesian Coordinates to Cylindrical Coordinates

Ex 2: Convert Cartesian Coordinates to Cylindrical Coordinates

Ex: Convert Cylindrical Coordinates to Cartesian Coordinates

Vector Valued Functions

Introduction to Vector Valued Functions

The Domain of a Vector Valued Function

Ex: Determine the Domain of a Vector Valued Function

Ex: Find the Point of Intersection of a Line Given by a Vector Function and a Coordinate Plane

Ex 1: Vector Valued Function - Curve of Intersection

Ex 2: Vector Valued Function - Curve of Intersection

Determining a Vector Valued Function from a Rectangular Equation

Determine a Vector Valued Function from the Intersection of Two Surfaces

Limits of Vector Valued Functions

The Derivative of a Vector Valued Function

Ex: Find a Tangent Vector of a Space Curve Given by a Vector Valued Function

Ex: Find the Velocity and Acceleration Vector Given the Position Vector Valued Function

Ex: Find the Velocity and Position Vector Functions Given the Acceleration Vector Function

Ex: Find Parametric Equations of a Tangent Line to a Space Curve

Ex: Find a Unit Tangent Vector to a Space Curve Given by a Vector Valued Function

Determining Where a Space Curve is Smooth from a Vector Valued Function

Indefinite Integration of Vector Valued Functions

Indefinite Integration of Vector Valued Functions with Initial Conditions

Definite Integration of Vector Valued Functions

Ex: Integrate a Vector Valued Function

Properties of the Derivatives of Vector Valued Functions

The Derivative of the Cross Product of Two Vector Valued Functions

Determining Velocity, Speed, and Acceleration Using a Vector Valued Function

Determining the Unit Tangent Vector

Determining the Unit Normal Vector

Proving the Unit Normal Vector Formula

Determining a Tangent Line of a Curve Defined by a Vector Valued Function

Determining the Tangential and Normal Components of Acceleration

Determining Arc Length of a Curve Defined by a Vector Valued Function

Ex: Determine Arc Length of a Spiral Given by Parametric Equations

Ex: Determine Arc Length of a Helix Given by a Vector Valued Function

Determining Curvature of a Curve Defined by a Vector Valued Function

Ex 1: Find the Curvature of a Space Curve Given by a Vector Function (Cross Product)

Ex 2A: Find the Curvature of a Space Curve Given by a Vector Function (Cross Product)

Ex 2B: Find the Curvature of a Space Curve Given by a Vector Function (No Cross Product)

Find the Angle of Intersection of Two Space Curves Given As Vector Functions

Determining the Binormal Vector

Proofs

Proof of the Fundamental Theorem of Calculus (Part 1)

Proof of the Fundamental Theorem of Calculus (Part 2)

Proof of the formula for the Angle Between Two Vectors

Vector Projection

Proof of the Vector Projection Formula