Course Overview

MAC 2312 Calculus II is a continuation of Calculus I, focusing on advanced integration techniques, applications of integration, infinite sequences and series, and the study of parametric and polar equations. The course emphasizes conceptual understanding, mathematical modeling, and real-world applications.

Course Topics and Structure

Integration Techniques and Applications

This section covers advanced methods for evaluating integrals and their applications in geometry and physics.

• Integration by Substitution: A method for simplifying integrals by changing variables.

• Integration by Parts: Based on the product rule for differentiation, useful for integrating products of functions. Formula:

• Trigonometric Integrals and Substitution: Techniques for integrating expressions involving trigonometric functions.

• Partial Fraction Decomposition: Breaking rational functions into simpler fractions for easier integration.

• Improper Integrals: Integrals with infinite limits or discontinuous integrands, evaluated as limits.

Applications:

• Areas Between Curves: Calculating the area enclosed by two or more curves. Formula:

• Volumes of Solids of Revolution: Using the disk, washer, and cylindrical shell methods to find volumes. Disk Method: Washer Method: Cylindrical Shells:

• Arc Length: Finding the length of a curve between two points. Formula:

• Work: Calculating work done by variable forces, including applications to springs and fluids.

Infinite Sequences and Series

This section introduces the concept of sequences and series, convergence tests, and power series representations of functions.

• Sequences: Ordered lists of numbers, often defined recursively or by a formula.

• Series: The sum of the terms of a sequence.

• Convergence Tests: Methods to determine if a series converges or diverges, including the integral test, comparison tests, ratio/root tests, and alternating series test.

• Power Series: Series of the form representing functions as infinite polynomials.

• Taylor and Maclaurin Series: Special power series centered at (Taylor) or (Maclaurin) for approximating functions. Formula:

Parametric Equations and Polar Coordinates

This section explores alternative ways to represent curves and compute areas and lengths in the plane.

• Parametric Equations: Representing curves by expressing and as functions of a parameter .

• Polar Coordinates: Describing points in the plane using radius and angle .

• Areas and Lengths in Polar Coordinates: Calculating areas and arc lengths for curves given in polar form. Area Formula:

Course Learning Outcomes

• Apply quantitative and logical skills to solve calculus problems.

• Represent and evaluate mathematical information numerically, graphically, and symbolically.

• Interpret mathematical models and draw inferences from them.

• Become proficient in techniques of integration and their applications.

• Understand and analyze the convergence of infinite series and sequences.

• Compute areas and volumes using calculus methods.

• Solve real-world problems involving calculus concepts from the physical sciences.

• Perform basic vector algebra (as relevant to the course).

• Understand and apply parametric equations and polar coordinates.

Assessment and Grading

Calculator Policy

• Basic scientific calculators (e.g., TI-30X, Casio fx-300 Es Plus) are allowed.

• Graphing calculators, smartphones, or similar devices are not permitted on assignments.

Course Schedule (Selected Topics by Week)

Success Tips

• Attend every class and participate actively in group activities.

• Complete assigned readings and homework before class.

• Review your work after class and practice additional exercises.

• Communicate with your instructor about any difficulties as soon as possible.

Instructor and Contact Information

• Instructor: Brian Johnson, Ph.D., Associate Professor

• Email: bpjohnson@fgcu.edu

• Office: Seidler Hall 213

• Office Hours: Mon, Wed, Fri 2-3pm; Thu 2:30-3:30pm; or by appointment

General Education Statement

This course is part of the General Education Program, designed to foster inquiry, creativity, and critical thinking, preparing students for academic and professional success.