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# Calculus II

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

## Content

### Integration

• 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.

### Differential Equations

• 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.

### Parametric/Polar Curves and Vector-Valued Functions

• 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.

### Infinite Sequences and Series

• 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. Surface Areas 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. 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
24 topics
3.13. Parametric Equations
 3.13.1. Differentiating Parametric Curves 3.13.2. Calculating Tangent and Normal Lines with Parametric Equations 3.13.3. Second Derivatives of Parametric Equations 3.13.4. Arc Lengths of Parametric Curves 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 Vector-Valued Functions 3.14.2. Differentiating Vector-Valued Functions 3.14.3. 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
46 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. Further Properties of Infinite Series 4.19.5. 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