BackMATH 205: Differential & Integral Calculus II – Course Syllabus and Topic Overview
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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 to .
Indefinite Integral: Represents the family of all antiderivatives of a function.
Notation: (definite), (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:
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
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.
