Further your understanding of calculus: master advanced integration techniques, model real-world situations using differential equations, and more.

- Use integration to compute arc length, area, and volume.
- Solve advanced integrals using techniques like substitution, long division, completing the square, partial fractions, and integration by parts.
- Evaluate improper integrals.

- Construct differential equations to model real-world situations involving rates of change.
- Verify solutions of differential equations, sketch slope fields, and relate solutions to slope fields.
- Estimate solutions of differential equations using Euler’s method.
- Find solutions of initial value problems using separation of variables.
- Interpret exponential and logistic growth models in context.

- Extend differentiation techniques to parametric and polar curves.
- Solve motion problems by differentiating and integrating vector-valued functions.
- Use integration to calculate the area of a region bounded by polar curves.

- Understand convergence and divergence in the context of sequences and series.
- Apply tests to determine the convergence or divergence of a series.
- Construct Taylor polynomials and use them to approximate values of functions.
- Compute and interpret the radius and interval of convergence of a power series.

1.

Integration Techniques
40 topics

1.1. Integration Using Substitution

1.1.1. | Integrating Algebraic Functions Using Substitution | |

1.1.2. | Integrating Linear Rational Functions Using Substitution | |

1.1.3. | Integration Using Substitution | |

1.1.4. | Calculating Definite Integrals Using Substitution | |

1.1.5. | Further Integration of Algebraic Functions Using Substitution | |

1.1.6. | Integrating Exponential Functions Using Linear Substitution | |

1.1.7. | Integrating Exponential Functions Using Substitution | |

1.1.8. | Integrating Trigonometric Functions Using Substitution | |

1.1.9. | Integrating Logarithmic Functions Using Substitution | |

1.1.10. | Integration by Substitution With Inverse Trigonometric Functions | |

1.1.11. | Integrating Hyperbolic Functions Using Substitution | |

1.1.12. | Integration by Substitution With Inverse Hyperbolic Functions | |

1.1.13. | Integration by Substitution With Inverse Reciprocal Hyperbolic Functions |

1.2. Special Techniques for Integration

1.2.1. | Integrating Functions Using Polynomial Division | |

1.2.2. | Integrating Functions by Completing the Square | |

1.2.3. | Integration Using Hyperbolic Functions and Completing the Square | |

1.2.4. | Integration Using Inverse Reciprocal Hyperbolic Functions and Completing the Square |

1.3. Integration Using Trigonometric Identities

1.3.1. | Integration Using Basic Trigonometric Identities | |

1.3.2. | Integration Using the Pythagorean Identities | |

1.3.3. | Integration Using the Double Angle Formulas |

1.4. Integration Using Hyperbolic Identities

1.4.1. | Integration Using Basic Hyperbolic Identities | |

1.4.2. | Integration Using the Hyperbolic Pythagorean Identities | |

1.4.3. | Integration Using the Hyperbolic Double Angle Formulas |

1.5. Integration by Parts

1.5.1. | Introduction to Integration by Parts | |

1.5.2. | Using Integration by Parts to Calculate Integrals With Logarithms | |

1.5.3. | Applying the Integration By Parts Formula Twice | |

1.5.4. | The Tabular Method of Integration by Parts | |

1.5.5. | Integration by Parts in Cyclic Cases |

1.6. Integration Using Partial Fractions

1.6.1. | Expressing Rational Functions as Sums of Partial Fractions | |

1.6.2. | Expressing Rational Functions with Repeated Factors as Sums of Partial Fractions | |

1.6.3. | Expressing Rational Functions with Irreducible Quadratic Factors as Sums of Partial Fractions | |

1.6.4. | Integrating Rational Functions Using Partial Fractions | |

1.6.5. | Integrating Rational Functions with Repeated Factors | |

1.6.6. | Integrating Rational Functions with Irreducible Quadratic Factors |

1.7. Improper Integrals

1.7.1. | Improper Integrals | |

1.7.2. | Improper Integrals Involving Exponential Functions | |

1.7.3. | Improper Integrals Involving Arctangent | |

1.7.4. | Improper Integrals Over the Real Line | |

1.7.5. | Improper Integrals of the Second Kind | |

1.7.6. | Improper Integrals of the Second Kind: Discontinuities at Interior Points |

2.

Applications of Integration
27 topics

2.8. Applications of Integration

2.8.1. | Integrating Rates of Change | |

2.8.2. | Integrating Density Functions | |

2.8.3. | The Average Value of a Function | |

2.8.4. | The Area Between Curves Expressed as Functions of X | |

2.8.5. | The Area Between Curves Expressed as Functions of Y | |

2.8.6. | Finding Areas Between Curves that Intersect at More Than Two Points | |

2.8.7. | The Arc Length of a Smooth Planar Curve |

2.9. Volumes of Solids With Known Cross Sections

2.9.1. | Volumes of Solids with Square Cross Sections | |

2.9.2. | Volumes of Solids with Rectangular Cross Sections | |

2.9.3. | Volumes of Solids with Triangular Cross Sections | |

2.9.4. | Volumes of Solids with Circular Cross Sections |

2.10. Volumes of Revolution

2.10.1. | Volumes of Revolution Using the Disc Method: Rotation About the Coordinate Axes | |

2.10.2. | Volumes of Revolution Using the Disc Method: Rotation About Other Axes | |

2.10.3. | Volumes of Revolution Using the Washer Method: Rotation About the Coordinate Axes | |

2.10.4. | Volumes of Revolution Using the Washer Method: Rotation About Other Axes | |

2.10.5. | The Shell Method: Rotating a Region About the X-Axis | |

2.10.6. | The Shell Method: Rotating a Region Between Two Curves About the X-Axis | |

2.10.7. | The Shell Method: Rotation About the Y-Axis |

2.11. Surface Areas of Revolution

2.11.1. | Surface Areas of Revolution: Rotation About the X-Axis | |

2.11.2. | Surface Areas of Revolution: Rotation About the Y-Axis | |

2.11.3. | Calculating Surface Areas of Revolution for Parametric Curves |

2.12. Connecting Position, Velocity and Acceleration Using Integrals

2.12.1. | Calculating Velocity Using Integration | |

2.12.2. | Determining Characteristics of Moving Objects Using Integration | |

2.12.3. | Calculating the Position Function of a Particle Using Integration | |

2.12.4. | Calculating the Displacement of a Particle Using Integration | |

2.12.5. | Calculating the Total Distance Traveled by a Particle | |

2.12.6. | Average Position, Velocity, and Acceleration |

3.

Parametric & Polar Equations
23 topics

3.13. Parametric Equations

3.13.1. | Defining and Differentiating Parametric Equations | |

3.13.2. | Calculating Tangent and Normal Lines with Parametric Equations | |

3.13.3. | Second Derivatives of Parametric Equations | |

3.13.4. | Finding the Arc Lengths of Curves Given by Parametric Equations | |

3.13.5. | Calculating Areas Bounded by Parametric Functions | |

3.13.6. | Further Calculating Areas Bounded by Parametric Functions |

3.14. Vector-Valued Functions

3.14.1. | Defining and Differentiating Vector-Valued Functions | |

3.14.2. | Integrating Vector-Valued Functions |

3.15. The Planar Motion of a Particle

3.15.1. | Calculating Velocity for Plane Motion Using Differentiation | |

3.15.2. | Calculating Acceleration for Plane Motion Using Differentiation | |

3.15.3. | Finding Velocity Vectors in Two Dimensions Using Integration | |

3.15.4. | Finding Displacement Vectors in Two Dimensions Using Integration | |

3.15.5. | Applying Newton's Second Law in the Coordinate Plane |

3.16. Polar Coordinates

3.16.1. | Differentiating Curves Given in Polar Form | |

3.16.2. | Further Differentiation of Curves Given in Polar Form | |

3.16.3. | Horizontal and Vertical Tangents to Polar Curves | |

3.16.4. | Horizontal and Vertical Tangents to Polar Curves in Non-Differentiable Cases | |

3.16.5. | Tangent and Normal Lines to Polar Curves | |

3.16.6. | Finding the Area of a Polar Region | |

3.16.7. | Finding the Limits of Integration For a Given Polar Region | |

3.16.8. | The Total Area Bounded by a Single Polar Curve | |

3.16.9. | The Area Bounded by Two Polar Curves | |

3.16.10. | The Arc Length of a Polar Curve |

4.

Sequences & Series
45 topics

4.17. Sequences

4.17.1. | Limits of Sequences | |

4.17.2. | Convergence of Geometric Sequences | |

4.17.3. | Further Convergence of Geometric Sequences | |

4.17.4. | Limits of Sequences With Factorials | |

4.17.5. | Determining Limits of Sequences Using Relative Magnitudes | |

4.17.6. | Further Determining Limits of Sequences Using Relative Magnitudes |

4.18. Monotonic Sequences

4.18.1. | Monotonic Sequences | |

4.18.2. | Identifying Monotonic Sequences Using Differentiation | |

4.18.3. | Identifying Monotonic Sequences Using Ratios |

4.19. Infinite Series

4.19.1. | Infinite Series and Partial Sums | |

4.19.2. | Convergent and Divergent Infinite Series | |

4.19.3. | Properties of Infinite Series | |

4.19.4. | Telescoping Series |

4.20. Geometric Series

4.20.1. | Finding the Sum of an Infinite Geometric Series | |

4.20.2. | Writing an Infinite Geometric Series in Sigma Notation | |

4.20.3. | Finding the Sum of an Infinite Geometric Series Expressed in Sigma Notation | |

4.20.4. | Convergence of Geometric Series | |

4.20.5. | Repeating Decimals as Infinite Geometric Series |

4.21. Infinite Series Convergence Tests

4.21.1. | The Nth Term Test for Divergence | |

4.21.2. | The Integral Test | |

4.21.3. | The Remainder Estimate for the Integral Test | |

4.21.4. | Harmonic Series and p-Series | |

4.21.5. | The Comparison Test | |

4.21.6. | The Limit Comparison Test | |

4.21.7. | The Alternating Series Test | |

4.21.8. | The Ratio Test | |

4.21.9. | The Root Test | |

4.21.10. | Absolute and Conditional Convergence | |

4.21.11. | The Alternating Series Error Bound | |

4.21.12. | Determining Convergence Parameters for Infinite Series | |

4.21.13. | Selecting Procedures for Analyzing Infinite Series |

4.22. Taylor Polynomials

4.22.1. | Second-Degree Taylor Polynomials | |

4.22.2. | Analyzing Second-Degree Taylor Polynomials | |

4.22.3. | Third-Degree Taylor Polynomials | |

4.22.4. | Higher-Degree Taylor Polynomials | |

4.22.5. | The Lagrange Error Bound |

4.23. Taylor Series

4.23.1. | Radius of Convergence of Power Series Centered at the Origin | |

4.23.2. | Radius of Convergence of Power Series | |

4.23.3. | Maclaurin Series | |

4.23.4. | Taylor Series | |

4.23.5. | Representing Functions as Power Series | |

4.23.6. | Recognizing Standard Maclaurin Series | |

4.23.7. | Recognizing Standard Maclaurin Series for Trigonometric Functions | |

4.23.8. | Differentiating Taylor Series | |

4.23.9. | Approximating Integrals Using Taylor Series |

5.

Differential Equations
23 topics

5.24. Introduction to Differential Equations

5.24.1. | Introduction to Differential Equations | |

5.24.2. | Verifying Solutions of Differential Equations | |

5.24.3. | Solving Differential Equations Using Direct Integration | |

5.24.4. | Solving First-Order ODEs Using Separation of Variables | |

5.24.5. | Solving Initial Value Problems Using Separation of Variables | |

5.24.6. | Modeling With Differential Equations | |

5.24.7. | Further Modeling With Differential Equations |

5.25. Qualitative Techniques for Differential Equations

5.25.1. | Qualitative Analysis of Differential Equations | |

5.25.2. | Equilibrium Solutions of Differential Equations |

5.26. Modeling Exponential Growth and Decay With Differential Equations

5.26.1. | Exponential Growth and Decay Models With Differential Equations | |

5.26.2. | Exponential Growth and Decay Models With Differential Equations: Calculating Unknown Times and Initial Values | |

5.26.3. | Exponential Growth and Decay Models With Differential Equations: Half-Life Problems |

5.27. Modeling Logistic Growth With Differential Equations

5.27.1. | Logistic Growth Models With Differential Equations | |

5.27.2. | Qualitative Analysis of the Logistic Growth Equation | |

5.27.3. | Solving the Logistic Growth Equation |

5.28. Slope Fields

5.28.1. | Slope Fields for Directly Integrable Differential Equations | |

5.28.2. | Slope Fields for Autonomous Differential Equations | |

5.28.3. | Slope Fields for Nonautonomous Differential Equations | |

5.28.4. | Analyzing Slope Fields for Directly Integrable Differential Equations | |

5.28.5. | Analyzing Slope Fields for Autonomous Differential Equations | |

5.28.6. | Analyzing Slope Fields for Nonautonomous Differential Equations |

5.29. Numerical Solutions of Differential Equations

5.29.1. | Euler's Method: Calculating One Step | |

5.29.2. | Euler's Method: Calculating Multiple Steps |