Concordia MATH 205 Calculus II: Syllabus, Topics and Practice
A study guide for Concordia MATH 205 (Differential and Integral Calculus II): integrals, the Fundamental Theorem, techniques, volumes and series.
Concordia University Updated 3 min read
At a glance
- Course code
- MATH 205
- Credits
- 3 (lecture and tutorial)
- Prerequisites
- MATH 203 or equivalent
- Note
- Students in BSc programs or the BA programs in Mathematics and Statistics may not count it toward their program of concentration
- Covers
- Definite integrals and antiderivatives, the Fundamental Theorem, techniques of integration, improper integrals, area between curves, volumes of revolution, sequences and series, convergence tests, power, Taylor and Maclaurin series
- Next course
- MAST 218 (Multivariable Calculus I), with MATH 204
- Practice here
- Calculus 2
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On this page
MATH 205 is Differential and Integral Calculus II at Concordia University, the second half of the MATH 203 and MATH 205 pair. Because MATH 203 is all about derivatives, MATH 205 carries the whole of integral calculus and finishes with infinite series. This guide maps every MATH 205 topic to free practice so you can build each skill with short problems and instant feedback.
What Concordia MATH 205 covers
The Concordia calendar lists definite integrals and antiderivatives, the Fundamental Theorem of Calculus, indefinite integrals and techniques of integration (substitutions, integration by parts, partial fractions, trigonometric functions), improper integrals, applications of integration (area between curves, volumes of solids of revolution), and infinite sequences and series: tests for convergence, power series, and Taylor and Maclaurin series. It is a 3-credit course with lectures and tutorials, and the prerequisite is MATH 203 or equivalent.
That is the integral half of a typical Calculus 1 plus most of a typical Calculus 2. Parametric and polar curves and differential equations are not in the calendar description.
MATH 205 topic map
| MATH 205 topic | Practice | Read |
|---|---|---|
| Definite integrals | Riemann sums, signed area | Riemann sums guide, definite integral properties |
| Antiderivatives | Undoing derivatives | Antiderivatives guide |
| The Fundamental Theorem | FTC part 1, FTC part 2 | FTC guide |
| Substitution | u-substitution | u-substitution guide |
| Integration by parts | Integration by parts | Integration by parts rule |
| Partial fractions | Partial fractions | Partial fractions guide |
| Trigonometric integrals | Trig integrals, trig substitution | Integral of sin squared x |
| Improper integrals | Infinite limits of integration | Improper integrals guide |
| Area between curves | Area between curves | Area between curves guide |
| Volumes of revolution | Disks, washers | Volumes of revolution |
| Sequences and convergence tests | Sequences, choosing a test | Convergence tests |
| Power, Taylor and Maclaurin series | Power series and radius, Maclaurin series | Taylor series guide |
MATH 205 in the Concordia sequence
| Course | Title | Main focus |
|---|---|---|
| MATH 203 | Differential and Integral Calculus I | Limits and derivatives |
| MATH 205 | Differential and Integral Calculus II | Integrals and series |
| MAST 218 | Multivariable Calculus I | Several variables, needs MATH 204 and 205 |
Example problems at the MATH 205 level
1. Washers. Rotate the region between y = x and y = x^2 about the x-axis:
2. A trig integral. Using \sin^2 x = \frac{1 - \cos 2x}{2}:
3. The alternating series test. \sum \frac{(-1)^n}{n} converges because \frac{1}{n} decreases to 0, even though \sum \frac{1}{n} diverges.
4. Radius of convergence. For \sum n!\,x^n, the ratio test gives (n + 1)|x| \to \infty unless x = 0, so R = 0.
How to do well in MATH 205
- Keep derivative rules sharp. Integration is differentiation in reverse, and weak rules from MATH 203 slow everything down.
- Practise mixed integrals. The hard part is choosing a method. Sets like choosing a strategy build that skill.
- Sketch areas and volumes. A picture shows which curve is on top and where the region starts and ends.
- State your test for each series. Write the test, check its conditions, then conclude. That is where marks are given.
Tip: when a series has factorials or powers like 3^n, try the ratio test first.
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Area with rectangles
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