Calculus II (MATH 242) Syllabus and Course Structure: Chapters 6–9 Study Guide

Course Overview

This course, Calculus II (MATH 242), covers advanced topics in single-variable calculus, focusing on integral calculus and infinite series. The course is structured around chapters 6 through 9 of the textbook University Calculus, 4th ed., by Hass, Weir, and Thomas. Students will develop both theoretical understanding and practical skills in integration, applications of integrals, techniques of integration, improper integrals, and infinite series.

Course Learning Objectives

• Set up and evaluate integral formulas for quantities such as volumes, arc lengths, surface areas, center of mass, and work.

• Find areas, volumes, and arc lengths using integration.

• Apply techniques of integration, including integration by parts and integration of rational functions by partial fractions.

• Understand the concept of improper integrals and apply tests of convergence.

• Understand and solve separable differential equations.

• Apply tests of convergence for series, including the comparison test, ratio test, and root test.

• Understand the concept and properties of power series, including Taylor and Maclaurin series.

• Find power series expansions for well-known functions and apply them.

Course Content by Chapter

Chapter 6: Applications of Definite Integrals

This chapter explores how definite integrals can be used to solve real-world problems involving areas, volumes, and other physical quantities.

• Setting up integrals for quantities such as area, volume, arc length, surface area, center of mass, and work.

• Finding areas between curves: $A = \int_a^b [f(x) - g(x)]\,dx$

• Volumes of solids of revolution using the disk and washer methods:

  • Disk method: $V = \pi \int_a^b [R(x)]^2\,dx$

  • Washer method: $V = \pi \int_a^b ([R(x)]^2 - [r(x)]^2)\,dx$

• Arc length of a curve $y = f(x)$ from $x = a$ to $x = b$:

  • $L = \int_a^b \sqrt{1 + [f'(x)]^2}\,dx$

• Surface area of a solid of revolution:

  • $S = 2\pi \int_a^b f(x) \sqrt{1 + [f'(x)]^2}\,dx$

• Center of mass and work applications.

Example: Find the volume of the solid obtained by rotating $y = x^2$ about the x-axis from $x = 0$ to $x = 1$.

• Using the disk method: $V = \pi \int_0^1 (x^2)^2 dx = \pi \int_0^1 x^4 dx = \pi \left[ \frac{x^5}{5} \right]_0^1 = \frac{\pi}{5}$

Chapter 7: Integrals and Transcendental Functions

This chapter introduces integration involving transcendental functions such as exponential, logarithmic, and trigonometric functions.

• Integration of exponential and logarithmic functions:

  • $\int e^{ax} dx = \frac{1}{a} e^{ax} + C$

  • $\int \frac{1}{x} dx = \ln|x| + C$

• Integration of trigonometric functions:

  • $\int \sin(ax) dx = -\frac{1}{a} \cos(ax) + C$

  • $\int \cos(ax) dx = \frac{1}{a} \sin(ax) + C$

• Inverse trigonometric functions and their integrals.

Example: $\int \frac{1}{1 + x^2} dx = \arctan(x) + C$

Chapter 8: Techniques of Integration

This chapter covers advanced methods for evaluating integrals that cannot be solved by basic techniques.

• Integration by parts:

  • Formula: $\int u\,dv = uv - \int v\,du$

• Integration of rational functions by partial fractions:

  • Decompose $\frac{P(x)}{Q(x)}$ into simpler fractions before integrating.

• Trigonometric integrals and substitutions.

• Improper integrals and convergence tests:

  • Improper integrals involve infinite limits or unbounded integrands.

  • Example: $\int_1^\infty \frac{1}{x^2} dx = \lim_{b \to \infty} \int_1^b \frac{1}{x^2} dx = 1$

Example: Evaluate $\int x e^x dx$ using integration by parts.

• Let $u = x$, $dv = e^x dx$; then $du = dx$, $v = e^x$.

• $\int x e^x dx = x e^x - \int e^x dx = x e^x - e^x + C$

Chapter 9: Infinite Sequences and Series

This chapter introduces the theory and application of infinite sequences and series, including convergence tests and power series.

• Definition of a sequence: An ordered list of numbers $\{a_n\}$.

• Definition of a series: The sum $\sum_{n=1}^\infty a_n$.

• Convergence of series: A series converges if the sequence of partial sums approaches a finite limit.

• Tests for convergence:

  • Comparison Test

  • Ratio Test: $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L$; if $L < 1$, the series converges.

  • Root Test: $\lim_{n \to \infty} \sqrt[n]{|a_n|} = L$; if $L < 1$, the series converges.

• Power series: $\sum_{n=0}^\infty c_n (x - a)^n$

• Taylor and Maclaurin series:

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

  • Maclaurin series: Taylor series at $a = 0$

• Power series expansions for common functions, e.g., $e^x$, $\sin x$, $\cos x$.

Example: The Maclaurin series for $e^x$ is $e^x = \sum_{n=0}^\infty \frac{x^n}{n!}$

Course Schedule Overview

Week	Topics / Activities
1–3	Ch. 6: Applications of Definite Integrals (Sections 6.1–6.6)
4–5	Review Ch. 6, Exam 1, Begin Ch. 7 & 8 (Sections 7.2, 8.1)
6–8	Ch. 8: Techniques of Integration (Sections 8.2–8.7), Exam 2
9–16	Ch. 9: Infinite Sequences and Series (Sections 9.1–9.9), Exams 3 & Final

Key Grading Components

• Assignments: 16%

• Quizzes: 5%

• Exams: 16% + 16% + 24% (lowest of first three dropped)

• Calculus Lab: 18%

• Attendance: 5%

• Daily Worksheets: Required for quantitative reasoning

Letter Grade Standard

Percentage	Letter Grade
95% – 100%	A
90% – 95%	A−
87% – 90%	B+
83% – 87%	B
80% – 83%	B−
75% – 80%	C+
65% – 75%	C
60% – 65%	C−
55% – 60%	D
0% – 55%	F

Additional Info

• Online homework and quizzes are managed through MyMathLab (MML).

• Attendance and daily engagement are required for success.

• Final exam is cumulative, covering all topics from chapters 6–9.

Additional info: The above content is inferred and expanded from the course syllabus and schedule, with academic context added for clarity and completeness.