Calculus II Syllabus and Course Structure Overview

Calculus II Course Overview

Course Description

This course, Calculus II (MATH-2414), covers advanced topics in calculus, including differentiation and integration of transcendental functions, parametric equations, polar coordinates, integration techniques, sequences and series, and improper integrals. It is designed for students who have completed Calculus I and aims to deepen understanding of integral calculus and its applications.

• Prerequisite: Completion of Calculus I (MATH 2413)

• Credits: 4

• Topics: Definite integrals, area, volume, work, physical applications, integration techniques, improper integrals, sequences, series, Taylor and Maclaurin series, polar coordinates.

State-Defined Learning Outcomes

• Definite Integrals: Solve problems involving area, volume, work, and other physical applications using definite integrals.

• Integration Techniques: Use substitution, integration by parts, trigonometric substitution, partial fractions, and tables of anti-derivatives to evaluate definite and indefinite integrals.

• Improper Integrals: Define and evaluate improper integrals using concepts of limits, convergence, and divergence.

• Sequences and Series: Determine convergence or divergence of sequences and series.

• Taylor and Maclaurin Series: Represent functions using Taylor and Maclaurin series and integrate functions not integrable by conventional methods.

• Polar Coordinates: Use polar coordinates to find areas, lengths of curves, and represent conic sections.

Texas Core Objectives

• Critical Thinking Skills: Creative thinking, innovation, inquiry, analysis, evaluation, and synthesis of information.

• Communication Skills: Effective development, interpretation, and expression of ideas through written, oral, and visual communication.

• Empirical and Quantitative Skills: Manipulation and analysis of numerical data or observable facts resulting in informed conclusions.

• Teamwork: Ability to consider different points of view and work effectively with others.

• Personal Responsibility: Connecting choices, actions, and consequences to ethical decision-making.

• Social Responsibility: Intercultural competence, civic responsibility, and engagement in communities.

Course Structure and Grading

Graded Work

Type	Weight	Notes
Quiz	2%	Entry Quiz; must answer all questions correctly; can retake as needed
Online Homework	28%	Assignments in each section via MyLab Math; late work allowed with 30% penalty
Exams	45%	4 exams (each 15%); lowest score dropped; must be completed on time
Final Exam	25%	Comprehensive; must be completed on time

Grade Breakdown

Grade	Range
A	90 - 100%
B	80 - 89%
C	70 - 79%
D	60 - 69%
F	0 - 59%

Course Schedule (Sample)

• Week 1: Sections 6.2–6.7 (Applications of Integration)

• Week 2: Sections 8.2–8.6, 8.9 (Integration Techniques)

• Week 3: Sections 10.1–10.8 (Sequences and Series)

• Week 4: Sections 11.1–11.4 (Power Series), 12.1–12.3 (Parametric and Polar Curves)

• Week 5: Final Exam

Exams: Four exams and one comprehensive final exam, all timed and must be completed in one session.

Key Calculus II Topics

Applications of Integration

Definite integrals are used to solve problems involving area, volume, work, and other physical applications. These applications often require setting up integrals based on geometric or physical principles.

• Area under a curve: $A = \int_a^b f(x) \, dx$

• Volume by rotation: $V = \int_a^b \pi [f(x)]^2 \, dx$ (disk method)

• Work: $W = \int_a^b F(x) \, dx$

Integration Techniques

Advanced integration methods are essential for evaluating complex integrals. These include substitution, integration by parts, trigonometric substitution, partial fractions, and using tables of anti-derivatives.

• Substitution: $\int f(g(x)) g'(x) \, dx = \int f(u) \, du$

• Integration by Parts: $\int u \, dv = u v - \int v \, du$

• Trigonometric Substitution: Used for integrals involving $\sqrt{a^2 - x^2}$, $\sqrt{a^2 + x^2}$, $\sqrt{x^2 - a^2}$

• Partial Fractions: Decompose rational functions for easier integration

Improper Integrals

Improper integrals involve infinite limits or integrands with infinite discontinuities. Their evaluation requires understanding limits, convergence, and divergence.

• Definition: $\int_a^\infty f(x) \, dx = \lim_{b \to \infty} \int_a^b f(x) \, dx$

• Convergence: The integral converges if the limit exists and is finite.

• Divergence: The integral diverges if the limit does not exist or is infinite.

Sequences and Series

Sequences and series are foundational for understanding convergence and divergence, as well as representing functions as infinite sums.

• Sequence: An ordered list of numbers, often defined by a formula $a_n$

• Series: The sum of terms in a sequence $\sum_{n=1}^\infty a_n$

• Convergence: A series converges if its partial sums approach a finite limit.

• Divergence: A series diverges if its partial sums do not approach a finite limit.

Taylor and Maclaurin Series

Taylor and Maclaurin series are used to represent functions as infinite sums of their derivatives at a point. These series are useful for approximating functions and integrating functions not integrable by conventional methods.

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

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

Polar Coordinates and Parametric Equations

Polar coordinates and parametric equations provide alternative ways to represent curves and calculate areas and lengths.

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

• Length of Curve (Parametric): $L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt$

Course Policies and Support

Calculator Policy

• Allowed: Scientific calculators (e.g., TI-30)

• Not Allowed: Programmable/graphing calculators, cell phones, AI software

Testing Policies

• Exams must be completed on time and in one session.

• Testing options include on-campus testing centers or online with LockDown Browser and external webcam.

• Strict rules for calculator and resource use during exams.

Attendance and Participation

• Online presence and participation are required for success.

• Efficient time management is essential due to the fast pace of online courses.

Support Resources

• Success Coach, free tutoring, student resources (counseling, child care, housing, emergency aid, food pantries), technical support.

Institutional Policies

• Accommodations for disabilities, class drop/repeat options, Title IX, and more are available through Dallas College policies.

Additional info: The syllabus outlines the structure and expectations for Calculus II, including coverage of chapters 6, 8, 10, 11, and 12, which correspond to applications of integration, integration techniques, sequences and series, power series, and parametric/polar curves. This aligns with standard Calculus II content.