MATH 205: Differential & Integral Calculus II – Course Syllabus and Topic Overview

Course Overview

MATH 205: Differential & Integral Calculus II is a continuation of introductory calculus, focusing on advanced integration techniques, applications of the definite integral, and infinite sequences and series. The course is designed for students who have completed Calculus I and aims to deepen understanding of integral calculus and its applications.

Course Structure and Resources

• Prerequisite: MATH 203 or equivalent Calculus I course.

• Textbook: Thomas' Calculus: Early Transcendentals, Single Variable.

• Online Systems: WeBWorK (assignments and practice), MyLab Math (e-text, exercises, videos).

• Tutorials: Weekly sessions for practice and review of arithmetic, algebra, and calculus topics.

• Math Help Centre: Drop-in support staffed by graduate students.

Grading Scheme

• Assignments: 10%

• Midterm Test: 30% (or 0% if missed; final exam weight increases to 90%)

• Final Exam: 60% or 90%

Note: There is no "100% final exam" option; assignments always count for 10%.

Course Topics and Schedule

The following is an overview of the main topics covered in MATH 205, with references to textbook sections and recommended problems for practice.

Class #	Section	Topic	Key Concepts
1	5.1, 5.2, 5.3	Area and Estimating with Finite Sums; Sigma Notation; The Definite Integral	Riemann sums Summation notation Definition and properties of the definite integral
2	4.8, 5.4	Antiderivatives; The Fundamental Theorem of Calculus	Finding antiderivatives Relationship between differentiation and integration Evaluating definite integrals using antiderivatives
3	5.5, 5.6	Indefinite Integrals & Substitution; Area Between Curves	Integration by substitution Calculating area between curves
4	8.1, 8.2	Basic Integration Formulas; Integration by Parts	Standard integrals Integration by parts formula: $\int u \, dv = uv - \int v \, du$
5	8.3, 8.4	Trigonometric Integrals; Trigonometric Substitution	Integrals involving $\sin$, $\cos$, $\tan$, etc. Substitution for integrals involving $\sqrt{a^2 - x^2}$, etc.
6	8.5, 6.1	Partial Fractions; Volumes by Cross-Sections (Disk/Washer Method)	Decomposing rational functions Volume formulas: $V = \pi \int_a^b [R(x)^2 - r(x)^2] dx$
7	8.8	Improper Integrals	Integrals with infinite limits or discontinuous integrands Convergence and divergence
8	10.1, 10.2	Sequences; Infinite Series	Definition and properties of sequences Convergence of series
9	10.3, 10.4	The Integral Test; The Comparison Tests	Tests for convergence of series Direct and limit comparison tests
10	10.5, 10.6	Absolute Convergence, Ratio and Root Tests; Alternating Series	Ratio test: $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$ Root test: $\lim_{n \to \infty} \sqrt[n]{|a_n|}$ Alternating series test
11	10.7, 10.8	Power Series; Taylor and Maclaurin Series	Representation of functions as power series Taylor series: $f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n$ Maclaurin series (special case $a=0$)
12		Review	Comprehensive review of all topics

Key Integration Techniques

• Substitution: Used to simplify integrals by changing variables.

• Integration by Parts: Useful for products of functions.

• Trigonometric Substitution: Applies to integrals involving square roots of quadratic expressions.

• Partial Fractions: Decomposes rational functions for easier integration.

• Improper Integrals: Evaluates integrals with infinite limits or discontinuities.

Applications of Integration

• Area Between Curves: $A = \int_a^b [f(x) - g(x)] dx$

• Volumes of Solids of Revolution: Disk/washer method for finding volumes.

Infinite Sequences and Series

• Convergence Tests: Integral, comparison, ratio, root, and alternating series tests.

• Power Series: Representation of functions as infinite sums.

• Taylor and Maclaurin Series: Approximating functions using polynomials.

Academic Integrity and Student Conduct

• Students must adhere to Concordia University's Academic Code of Conduct and Code of Rights and Responsibilities.

• All course materials are the intellectual property of the instructor and may not be distributed without permission.

• Official communication is through Concordia email accounts.

Support and Resources

• Math Help Centre and tutorials provide additional support.

• WeBWorK and MyLab Math offer practice problems and resources.

• Student services are available for academic and personal support.