MATH 112: Calculus II – Syllabus and Course Outline Study Guide

Course Overview

MATH 112: Calculus II is a continuation of the calculus sequence, focusing on advanced integration techniques, applications of integration, infinite sequences and series, transcendental functions, and alternative coordinate systems such as polar coordinates. The course is designed to deepen students' understanding of calculus and its applications in science, engineering, and technology.

Course Topics and Structure

Integration and Its Applications

This section covers advanced methods of integration and their use in solving real-world problems.

• Volumes Using Cross Sections (Section 6.1): Techniques for finding the volume of solids by integrating the area of cross sections perpendicular to an axis.

• Volumes Using Cylindrical Shells (Section 6.2): Method for finding volumes by integrating cylindrical shells.

• Arc Length (Section 6.3): Calculating the length of a curve using integration.

• Areas of Surfaces of Revolution (Section 6.4): Finding the surface area generated by revolving a curve about an axis.

• Work (Section 6.5): Applying integration to compute work done by a variable force.

Key Formulas:

• Volume by Cross Sections: $V = \int_a^b A(x)\,dx$

• Volume by Cylindrical Shells: $V = 2\pi \int_a^b r(x)h(x)\,dx$

• Arc Length: $L = \int_a^b \sqrt{1 + [f'(x)]^2}\,dx$

• Surface Area of Revolution (about x-axis): $S = 2\pi \int_a^b f(x)\sqrt{1 + [f'(x)]^2}\,dx$

• Work: $W = \int_a^b F(x)\,dx$

Integration Techniques

Students learn advanced methods for evaluating integrals, including those involving transcendental functions.

• Integration by Parts (Section 8.2): $\int u\,dv = uv - \int v\,du$

• Trigonometric Integrals and Substitution (Sections 8.3, 8.4): Techniques for integrating products and powers of trigonometric functions.

• Integration of Rational Functions by Partial Fractions (Section 8.5): Decomposing rational expressions for easier integration.

• Numerical Integration (Section 8.7): Approximating definite integrals using methods such as the Trapezoidal Rule and Simpson's Rule.

• Improper Integrals (Section 8.8): Evaluating integrals with infinite limits or discontinuous integrands.

Example: To integrate $\int x e^x\,dx$, use integration by parts with $u = x$, $dv = e^x dx$.

Transcendental and Hyperbolic Functions

Exploration of exponential, logarithmic, and hyperbolic functions, including their properties and integrals.

• Exponential and Logarithmic Functions: Integration and differentiation of $e^x$, $\ln x$, and related functions.

• Hyperbolic Functions (Section 7.3): Definitions and calculus involving $\sinh x$, $\cosh x$, etc.

Key Formulas:

• $\frac{d}{dx} e^x = e^x$

• $\frac{d}{dx} \ln x = \frac{1}{x}$

• $\sinh x = \frac{e^x - e^{-x}}{2}$, $\cosh x = \frac{e^x + e^{-x}}{2}$

Infinite Sequences and Series

This topic introduces sequences, series, and their convergence properties, including tests for convergence and power series representations.

• Sequences (Section 10.1): Ordered lists of numbers, convergence, and limits.

• Infinite Series (Section 10.2): Sums of infinite sequences, convergence tests.

• Integral Test (Section 10.3): Using integrals to determine series convergence.

• Comparison, Ratio, and Root Tests (Sections 10.4, 10.5): Methods for testing series convergence.

• Alternating Series, Absolute and Conditional Convergence (Section 10.6): Series with alternating terms and their convergence properties.

• Power Series (Section 10.7): Representing functions as infinite sums of powers.

• Taylor and Maclaurin Series (Section 10.8): Approximating functions with polynomials.

• Convergence of Taylor Series (Section 10.9): Determining where Taylor series converge to the function.

• Binomial Series (Section 10.10): Series expansion for $(1 + x)^n$.

Key Formulas:

• Geometric Series: $\sum_{n=0}^\infty ar^n = \frac{a}{1 - r}$, $|r| < 1$

• Taylor Series: $f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x - a)^n$

Parametric Equations and Polar Coordinates

Students learn alternative ways to represent curves and compute calculus quantities in these systems.

• Parametrizations of Plane Curves (Section 11.1): Expressing curves as $x = f(t)$, $y = g(t)$.

• Calculus with Parametric Curves (Section 11.2): Differentiation and integration for parametric equations.

• Polar Coordinates (Section 11.3): Representing points as $(r, \theta)$ and graphing polar equations.

• Areas and Lengths in Polar Coordinates (Section 11.5): Calculating area and arc length in polar form.

Key Formulas:

• Area in Polar Coordinates: $A = \frac{1}{2} \int_{\alpha}^{\beta} [r(\theta)]^2 d\theta$

• Arc Length in Polar Coordinates: $L = \int_{\alpha}^{\beta} \sqrt{[r(\theta)]^2 + [r'(\theta)]^2} d\theta$

Course Policies and Assessment

Grading Breakdown

Component	Weight
Quizzes and Homework	17%
Common Midterm 1	17%
Common Midterm 2	17%
Common Midterm 3	17%
Final Exam	32%

Letter Grade Scale

Letter Grade	Score Range
A	88 - 100
B+	83 - 87
B	77 - 82
C+	72 - 76
C	66 - 71
D	60 - 65
F	0 - 59

Important Policies

• Attendance: Mandatory for all classes.

• Homework: Completed online via MyMathLab.

• Quizzes: Weekly, based on lectures and homework.

• Exams: Three common midterms and one comprehensive final exam.

• Academic Integrity: Strictly enforced; violations result in disciplinary action.

• AI Policy: Use of AI tools is prohibited for assignments and exams.

Course Schedule (Selected Topics)

Lecture	Section	Topic
1-2	6.1	Volumes Using Cross Sections
3	6.2	Volumes Using Cylindrical Shells
4	6.3	Arc Length
5	6.4	Areas of Surfaces of Revolution
6	6.5	Work
7	7.3	Hyperbolic Functions
8-17	8.1-8.8	Integration Techniques, Numerical and Improper Integrals
18-33	10.1-10.10	Sequences, Series, Convergence Tests, Power and Taylor Series
34-41	11.1-11.5	Parametric Equations, Polar Coordinates, Areas and Lengths

Additional info: The course outline follows the standard Calculus II curriculum, covering chapters 6, 7, 8, 10, and 11 from Thomas' Calculus: Early Transcendentals, 15th Edition.

Resources and Support

• Textbook: Thomas' Calculus: Early Transcendentals, 15th Edition by Hass, Heil, and Weir.

• Math Tutoring Center: Central King Building, Lower Level, Room G11.

• Office Hours: Check the Math Department's webpage for instructor office hours and emails.

• Accessibility: Contact the Office of Accessibility Resources and Services (OARS) for accommodations.

Key Dates

Date	Event
Jan 20, 2026	First Day of Classes
Feb 11, 2026	Midterm Exam 1
Mar 11, 2026	Midterm Exam 2
Apr 6, 2026	Last Day to Withdraw
Apr 22, 2026	Midterm Exam 3
May 8-14, 2026	Final Exam Period

Summary

This course provides a comprehensive study of advanced calculus topics, including integration techniques, applications, infinite series, and alternative coordinate systems. Mastery of these topics is essential for further study in mathematics, science, and engineering.