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 additional practice and review of arithmetic and algebra.

• 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% (depending on midterm participation)

Note: There is no 100% final exam option.

Schedule of Topics

The following topics are covered, corresponding to chapters in a standard calculus sequence:

Class	Section	Topic
1	5.1, 5.2, 5.3	Area and Estimating with Finite Sums; Sigma Notation and Limits of Finite Sums; The Definite Integral
2	4.8, 5.4	Anti-derivatives; The Fundamental Theorem of Calculus
3	5.5, 5.6	Indefinite Integrals & Substitution Method; Definite Integral Substitutions, Area Between Curves
4	8.1, 8.2	Using Basic Integration Formulas; Integration by Parts
5	8.3, 8.4	Trigonometric Integrals; Trigonometric Substitution
6	8.5, 6.1	Integration by Partial Fractions; Volumes Using Cross-Sections (Disk/Washer Method)
7	8.8	Improper Integrals
8	10.1, 10.2	Sequences; Infinite Series
9	10.3, 10.4	The Integral Test; The Comparison Tests
10	10.5, 10.6	Absolute Convergence, Ratio and Root Tests; Alternating Series & Conditional Convergence
11	10.7, 10.8	Power Series; Taylor and Maclaurin Series
12		Review Class

Key Topics and Concepts

Definite and Indefinite Integrals

• Definite Integral: Represents the signed area under a curve from $a$ to $b$.

• Indefinite Integral: Represents the family of all antiderivatives of a function.

• Notation: $\int_a^b f(x)\,dx$ (definite), $\int f(x)\,dx$ (indefinite)

• Fundamental Theorem of Calculus: Connects differentiation and integration.

Techniques of Integration

• Substitution Method: Useful for integrals involving composite functions.

• Integration by Parts: Based on the product rule for differentiation.

• Trigonometric Integrals and Substitution: For integrals involving trigonometric functions.

• Partial Fractions: Decomposes rational functions for easier integration.

Applications of Integrals

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

• Volumes by Cross-Sections: Disk/Washer method for solids of revolution.

• Improper Integrals: Integrals with infinite limits or discontinuous integrands.

Infinite Sequences and Series

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

• Series: Sums of sequences, including geometric and telescoping series.

• Convergence Tests: Integral, Comparison, Ratio, Root, and Alternating Series Tests.

• Power Series: Series of the form $\sum_{n=0}^\infty a_n(x-c)^n$

• Taylor and Maclaurin Series: Polynomial approximations of functions.

Academic Integrity and Conduct

• Students must adhere to Concordia's Academic Code of Conduct and policies on academic integrity.

• Respectful and professional behavior is expected in all course-related activities.

• Intellectual property rights apply to all course materials.

• Official communication must use Concordia email accounts.

Student Support

• Access to Math Help Centre, tutorials, and online resources is strongly encouraged.

• Additional student services are available through the university website.