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Notes for Math 220,
Calculus 2
Brenton LeMesurier
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\(\newcommand{\sech}{\textrm{sech}} \newcommand{\csch}{\text{csch}} \newcommand{\reals}{\mathbb{R}} \let\ds\displaystyle \newcommand{\integral}[3][]{\int#1 #2 \;d#3} \newcommand{\dsintegral}[3][]{\ds\int#1 #2 \;d#3} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \definecolor{fillinmathshade}{gray}{0.9} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} \)
Front Matter
1
Integration (review from first semester calculus)
1.1
Approximating Areas (and Distance Traveled)
1.2
The Definite Integral
1.3
The Fundamental Theorem of Calculus
1.4
Integration Formulas and the Net Change Theorem
1.5
Substitution
1.6
Integrals Involving Exponential and Logarithmic Functions — Summary
1.7
Integrals Resulting in Inverse Trigonometric Functions — Summary
2
Applications of Integration
2.1
Areas between Curves
2.2
Determining Volumes by Slicing
2.3
Volumes of Revolution: Cylindrical Shells
2.4
Arc Length of a Curve and Surface Area
2.5
Physical Applications (omitted)
2.6
Moments and Centers of Mass (omitted)
2.7
Integrals, Exponential Functions, and Logarithms (omitted)
2.8
Exponential Growth and Decay (omitted)
2.9
Hyperbolic Functions
2.9.1
Definitions
2.9.2
Identities
2.9.3
Derivatives of the Hyperbolic Functions
2.9.4
Domains, Ranges and Invertibility
2.9.5
Derivatives of the Inverse Hyperbolic Functions
2.9.6
Some Useful Integrals
3
Techniques of Integration
3.1
Integration by Parts
3.2
Trigonometric Integrals
3.3
Trigonometric Substitution
3.4
Partial Fractions
3.5
Other Strategies for Integration
3.6
Numerical Integration
3.7
Improper Integrals
4
Introduction to Differential Equations
4.1
Basics of Differential Equations
4.2
Direction Fields and Numerical Methods (omitted)
4.3
Separable Equations
4.4
The Logistic Equation (a very brief introduction)
4.5
First-order Linear Equations
5
Sequences and Series
5.1
Sequences
5.2
Infinite Series
5.3
The Divergence and Integral Tests
5.3.1
The Divergence Test
5.3.2
The Integral Test and Estimates of Sums
5.4
Comparison Tests
5.5
Alternating Series (and Conditional vs. Absolute Convergence)
5.6
Ratio and Root Tests
6
Power Series
6.1
Power Series and Functions
6.2
Properties of Power Series
6.3
Taylor and Maclaurin Series
6.4
Working with Taylor Series
7
Parametric Equations and Polar Coordinates
7.1
Parametric Equations
7.2
Calculus of Parametric Curves
7.3
Polar Coordinates
7.4
Area and Arc Length in Polar Coordinates
Appendices
A
Rules for Derivatives and Integrals
A.1
Rules for Derivatives
A.2
Rules for Integrals
B
Calculus Formula Checklists
B.1
Derivatives Checklist
B.2
Integrals Checklist
C
Reduction Formulas For Integrals
C.1
Integrals Involving Exponential or Trigonometric Functions
C.2
Integrals Involving Inverse Trigonometric Functions
C.3
Integrals Involving
\(\sqrt{a + bu}\)
D
Some Power Series
E
Some Trigonometry
F
Strategy for Evaluating Integrals
F.1
A few general tactics
F.2
A detailed strategy
F.2.1
Use tables of integrals and known integrals
F.2.2
Do basic simplifications
F.2.3
Substitution
F.2.4
Choosing a substitution function
\(u(x)\)
F.2.5
Integration by Parts
F.2.6
Inverse Substitution, especially with trigonometric functions
F.2.7
Special simplifications and substitutions for products of trigonometric functions
F.2.8
Integration of rational functions (ratios of polynomials)
F.2.9
Final steps: make sure that you answer the original question
G
Some Formulas Worth Knowing
Section
2.5
Physical Applications (omitted)
References.
OpenStax Calculus Volume 2, Section 2.5
1
.
openstax.org/books/calculus-volume-2/pages/2-5-physical-applications
