# Calculus II

## Course Overview

Calculus II is a comprehensive calculus course that covers advanced applications and techniques of integration, differential equations, and sequences and series. The course is intended as the second part of a sequence that begins with Calculus I.

Integral calculus emphasizes techniques of integration and its applications. Once students have shown competence with both integral and differential calculus, they are prepared for the final two concepts of the course: differential equations and sequences and series. The differential equations unit is designed to provide students with an introduction to the topic, which will give them an advantage when they take a more thorough differential equations course. The unit on sequences and series focuses primarily on tests for convergence and divergence, power series, expressing functions as power series, and Taylor/Maclaurin series.

## Course Content

### Unit 1: Applications of Integrals

Much like differential calculus, students learn integral calculus so they can use techniques to solve real-world problems. This unit does not use advanced techniques of integration, which are learned in Unit 6. This unit covers the following topics:

• Areas between curves
• Volumes of solids of revolution using the disk/washer method
• Volumes of solids of known cross section
• Volumes of solids of revolution using the shell method

### Unit 2: More Applications of Integrals and Integration by Parts

This unit concentrates on more advanced techniques of integration. This unit covers the following topics:

• Average Value of a Function
• Work
• Integration by parts

### Unit 3: Advanced Techniques of Integration

This unit concentrates on more advanced techniques of integration. This unit covers the following topics:

• Trigonometric Integrals
• Trigonometric substitution
• Partial fraction decomposition
• Strategies of integration

### Unit 4: More Advanced Techniques of Integration and Arc Length

This is the final unit in the course and can be the most challenging for students, as it if probably the most theoretical. The unit explores all aspects of convergence and divergence of sequences and series, and concludes with writing functions as power series using Taylor and Maclaurin polynomials. The unit covers the following topics:

• Integration using tables and calculators
• Numerical approximation of integration
• Improper integrals
• Arc length of curves

### Unit 5: Review of Differential Equations

This unit rounds out calculus of two-dimensional spaces, introducing students to using calculus on less traditional functions such as parametric and polar functions. The unit also includes and introduction to differential equations. The following topics are included in the unit:

• The definition of a differential equation
• Direction/slope fields and Euler’s Method
• Exponential growth and decay
• Separable differential equations
• Models for population growth

### Unit 6: Calculus of Polar and Parametric Equations and Differential Equations

This unit rounds out calculus of two-dimensional spaces, introducing students to using calculus on less traditional functions such as parametric and polar functions. The unit also includes and introduction to differential equations. The following topics are included in the unit:

• Parametric equations
• Calculus of parametric equations and curves
• Polar coordinates and graphs
• Calculus of polar equations

### Unit 7: Sequences and Series

This is the final unit in the course and can be the most challenging for students, as it if probably the most theoretical. The unit explores all aspects of convergence and divergence of sequences and series, and concludes with writing functions as power series using Taylor and Maclaurin polynomials. The unit covers the following topics:

• Sequences
• Series
• The integral test
• The comparison and limit comparison tests
• Alternating series
• The ratio and root test
• Practice determining convergence and divergence of series
• Power series
• Representing functions as power series
• Taylor and Maclaurin series
• Lagrange error bounds