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Calculus Volume 2

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Calculus Volume 2Index
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  1. Preface
  2. 1 Integration
    1. Introduction
    2. 1.1 Approximating Areas
    3. 1.2 The Definite Integral
    4. 1.3 The Fundamental Theorem of Calculus
    5. 1.4 Integration Formulas and the Net Change Theorem
    6. 1.5 Substitution
    7. 1.6 Integrals Involving Exponential and Logarithmic Functions
    8. 1.7 Integrals Resulting in Inverse Trigonometric Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  3. 2 Applications of Integration
    1. Introduction
    2. 2.1 Areas between Curves
    3. 2.2 Determining Volumes by Slicing
    4. 2.3 Volumes of Revolution: Cylindrical Shells
    5. 2.4 Arc Length of a Curve and Surface Area
    6. 2.5 Physical Applications
    7. 2.6 Moments and Centers of Mass
    8. 2.7 Integrals, Exponential Functions, and Logarithms
    9. 2.8 Exponential Growth and Decay
    10. 2.9 Calculus of the Hyperbolic Functions
    11. Key Terms
    12. Key Equations
    13. Key Concepts
    14. Chapter Review Exercises
  4. 3 Techniques of Integration
    1. Introduction
    2. 3.1 Integration by Parts
    3. 3.2 Trigonometric Integrals
    4. 3.3 Trigonometric Substitution
    5. 3.4 Partial Fractions
    6. 3.5 Other Strategies for Integration
    7. 3.6 Numerical Integration
    8. 3.7 Improper Integrals
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  5. 4 Introduction to Differential Equations
    1. Introduction
    2. 4.1 Basics of Differential Equations
    3. 4.2 Direction Fields and Numerical Methods
    4. 4.3 Separable Equations
    5. 4.4 The Logistic Equation
    6. 4.5 First-order Linear Equations
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter Review Exercises
  6. 5 Sequences and Series
    1. Introduction
    2. 5.1 Sequences
    3. 5.2 Infinite Series
    4. 5.3 The Divergence and Integral Tests
    5. 5.4 Comparison Tests
    6. 5.5 Alternating Series
    7. 5.6 Ratio and Root Tests
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Chapter Review Exercises
  7. 6 Power Series
    1. Introduction
    2. 6.1 Power Series and Functions
    3. 6.2 Properties of Power Series
    4. 6.3 Taylor and Maclaurin Series
    5. 6.4 Working with Taylor Series
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Chapter Review Exercises
  8. 7 Parametric Equations and Polar Coordinates
    1. Introduction
    2. 7.1 Parametric Equations
    3. 7.2 Calculus of Parametric Curves
    4. 7.3 Polar Coordinates
    5. 7.4 Area and Arc Length in Polar Coordinates
    6. 7.5 Conic Sections
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter Review Exercises
  9. A | Table of Integrals
  10. B | Table of Derivatives
  11. C | Review of Pre-Calculus
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
  13. Index
A
absolute convergence5.5 Alternating Series
absolute error3.6 Numerical Integration
air resistance4.5 First-order Linear Equations
Airy’s equation6.4 Working with Taylor Series
algebraic function3.1 Integration by Parts
alternating series5.5 Alternating Series
alternating series test5.5 Alternating Series
angular coordinate7.3 Polar Coordinates
annuities6.2 Properties of Power Series
annuity payments6.4 Working with Taylor Series
aphelion1.3 The Fundamental Theorem of Calculus
arc length2.4 Arc Length of a Curve and Surface Area
Archimedean spiral7.3 Polar Coordinates
Archimedes1.1 Approximating Areas
area density2.5 Physical Applications
area under the curve1.1 Approximating Areas
arithmetic sequence5.1 Sequences
asymptotically semi-stable solution4.2 Direction Fields and Numerical Methods
asymptotically stable solution4.2 Direction Fields and Numerical Methods
asymptotically unstable solution4.2 Direction Fields and Numerical Methods
autonomous differential equation4.3 Separable Equations
average value of the function1.2 The Definite Integral
B
bald eagle1.4 Integration Formulas and the Net Change Theorem
binomial series6.4 Working with Taylor Series
bounded above5.1 Sequences
bounded below5.1 Sequences
bounded sequence5.1 Sequences
C
carbon dating2.8 Exponential Growth and Decay
cardioid7.3 Polar Coordinates
carrying capacity4.4 The Logistic Equation
catenary2.9 Calculus of the Hyperbolic Functions
center of mass2.6 Moments and Centers of Mass
centroid2.6 Moments and Centers of Mass
chambered nautilusIntroduction, 7.3 Polar Coordinates
change of variables1.5 Substitution
cissoid of Diocles7.4 Area and Arc Length in Polar Coordinates
comparison test5.4 Comparison Tests
compound interest2.8 Exponential Growth and Decay
computer algebra systems (CAS)3.5 Other Strategies for Integration
conditional convergence5.5 Alternating Series
conic section7.5 Conic Sections
convergence of a series5.2 Infinite Series
convergent sequence5.1 Sequences
coupon collector’s problem5.3 The Divergence and Integral Tests
cross-section2.2 Determining Volumes by Slicing
curtate cycloid7.1 Parametric Equations
cusps7.1 Parametric Equations
cycloid7.1 Parametric Equations
D
deceleration1.4 Integration Formulas and the Net Change Theorem
definite integral1.2 The Definite Integral
density function2.5 Physical Applications
differential equation4.1 Basics of Differential Equations
direction field (slope field)4.2 Direction Fields and Numerical Methods
directrix7.5 Conic Sections
discriminant7.5 Conic Sections
disease epidemics4.2 Direction Fields and Numerical Methods
disk method2.2 Determining Volumes by Slicing
displacement1.2 The Definite Integral, 1.4 Integration Formulas and the Net Change Theorem
divergence of a series5.2 Infinite Series
divergence test5.3 The Divergence and Integral Tests
divergent sequence5.1 Sequences
doubling time2.8 Exponential Growth and Decay
drugs in the bloodstream4.3 Separable Equations
dummy variable1.1 Approximating Areas, 1.2 The Definite Integral
E
Earth’s orbit7.1 Parametric Equations
eccentricity7.5 Conic Sections
elliptic integral6.4 Working with Taylor Series
epitrochoid7.1 Parametric Equations
equilibrium solution4.2 Direction Fields and Numerical Methods
Euler transform5.5 Alternating Series
Euler’s constant5.2 Infinite Series
Euler’s formula6.4 Working with Taylor Series
Euler’s Method4.2 Direction Fields and Numerical Methods
evaluation theorem1.3 The Fundamental Theorem of Calculus
even function1.4 Integration Formulas and the Net Change Theorem
explicit formulas5.1 Sequences
exponential decay2.8 Exponential Growth and Decay
exponential growth2.8 Exponential Growth and Decay
F
fave1.2 The Definite Integral
federal income tax1.4 Integration Formulas and the Net Change Theorem
Fibonacci numbers5.1 Sequences
focal parameter7.5 Conic Sections
focus7.5 Conic Sections
Fresnel integrals6.4 Working with Taylor Series
fruit flies1.6 Integrals Involving Exponential and Logarithmic Functions
frustum2.4 Arc Length of a Curve and Surface Area
Fundamental Theorem of Calculus1.3 The Fundamental Theorem of Calculus
Fundamental Theorem of Calculus, Part 11.3 The Fundamental Theorem of Calculus
Fundamental Theorem of Calculus, Part 21.3 The Fundamental Theorem of Calculus
G
Gabriel’s Horn3.7 Improper Integrals
general form7.5 Conic Sections
general solution4.1 Basics of Differential Equations
geometric sequence5.1 Sequences
geometric series5.2 Infinite Series
golden ratio5.1 Sequences
Gompertz equation4.4 The Logistic Equation
growth of bacteria1.6 Integrals Involving Exponential and Logarithmic Functions
growth rate4.4 The Logistic Equation
H
half-life2.8 Exponential Growth and Decay
hanging cables2.9 Calculus of the Hyperbolic Functions
harmonic series5.2 Infinite Series
Hooke’s law2.5 Physical Applications
Hoover Dam2.5 Physical Applications
hydrostatic pressure2.5 Physical Applications
hypocycloid7.1 Parametric Equations
I
iceboat1.4 Integration Formulas and the Net Change Theorem
improper integral3.7 Improper Integrals
indefinite integrals3.1 Integration by Parts
index1.1 Approximating Areas
index variable5.1 Sequences
infinite sequence5.1 Sequences
infinite series5.2 Infinite Series
initial population4.4 The Logistic Equation
initial value4.1 Basics of Differential Equations
initial velocity4.1 Basics of Differential Equations
initial-value problem4.1 Basics of Differential Equations
integrable function1.2 The Definite Integral
integral test5.3 The Divergence and Integral Tests
integrand1.2 The Definite Integral
integrating factor4.5 First-order Linear Equations
integration by parts3.1 Integration by Parts
integration by substitution1.5 Substitution
integration tables3.5 Other Strategies for Integration
interval of convergence6.1 Power Series and Functions
J
joule2.5 Physical Applications
K
Koch’s snowflake5.2 Infinite Series
L
lamina2.6 Moments and Centers of Mass
Laplace transform3.7 Improper Integrals, 3.7 Improper Integrals
left-endpoint approximation1.1 Approximating Areas
Leibniz1.2 The Definite Integral
limaçon7.3 Polar Coordinates
limit comparison test5.4 Comparison Tests
limit of the sequence5.1 Sequences
limits of integration1.2 The Definite Integral
linear4.5 First-order Linear Equations
logarithmic function3.1 Integration by Parts
logistic differential equation4.4 The Logistic Equation
lower sum1.1 Approximating Areas
M
Maclaurin polynomials6.3 Taylor and Maclaurin Series
Maclaurin series6.3 Taylor and Maclaurin Series
major axis7.5 Conic Sections
Mean Value Theorem for Integrals1.3 The Fundamental Theorem of Calculus
method of cylindrical shells.2.3 Volumes of Revolution: Cylindrical Shells
method of equating coefficients3.4 Partial Fractions
method of exhaustion1.1 Approximating Areas
method of strategic substitution3.4 Partial Fractions
midpoint rule3.6 Numerical Integration
minor axis7.5 Conic Sections
moment2.6 Moments and Centers of Mass
monotone sequence5.1 Sequences
N
nappes7.5 Conic Sections
net change theorem1.4 Integration Formulas and the Net Change Theorem
net signed area1.2 The Definite Integral
Newton1.3 The Fundamental Theorem of Calculus
Newton’s law of cooling2.8 Exponential Growth and Decay, 4.3 Separable Equations
Newton’s second law of motion4.1 Basics of Differential Equations
nonelementary integral6.4 Working with Taylor Series
numerical integration3.6 Numerical Integration
O
odd function1.4 Integration Formulas and the Net Change Theorem
order of a differential equation4.1 Basics of Differential Equations
orientation7.1 Parametric Equations
P
p-series5.3 The Divergence and Integral Tests
parameter7.1 Parametric Equations
parameterization of a curve7.1 Parametric Equations
parametric curve7.1 Parametric Equations
parametric equations7.1 Parametric Equations
partial fraction decomposition3.4 Partial Fractions
partial sum5.2 Infinite Series
particular solution4.1 Basics of Differential Equations
partition1.1 Approximating Areas
Pascal’s principle2.5 Physical Applications
pascals2.5 Physical Applications
perihelion1.3 The Fundamental Theorem of Calculus
phase line4.4 The Logistic Equation
polar axis7.3 Polar Coordinates
polar coordinate system7.3 Polar Coordinates
polar equations7.3 Polar Coordinates
pole7.3 Polar Coordinates
Population growth2.8 Exponential Growth and Decay
power reduction formulas3.2 Trigonometric Integrals
power series6.1 Power Series and Functions
power-reducing identities3.2 Trigonometric Integrals
present value6.2 Properties of Power Series
price–demand function1.6 Integrals Involving Exponential and Logarithmic Functions
probability3.7 Improper Integrals
probability density function3.7 Improper Integrals
prolate cycloid7.1 Parametric Equations
R
radial coordinate7.3 Polar Coordinates
radial density2.5 Physical Applications
radius of convergence6.1 Power Series and Functions
Ramanujan5.6 Ratio and Root Tests
rate of change1.4 Integration Formulas and the Net Change Theorem
ratio test5.6 Ratio and Root Tests
rational functions3.4 Partial Fractions
RC circuit4.5 First-order Linear Equations
recurrence relation5.1 Sequences, 5.1 Sequences
regular partition1.1 Approximating Areas
relative error3.6 Numerical Integration
remainder estimate5.3 The Divergence and Integral Tests
Riemann sum1.1 Approximating Areas
Riemann sums3.6 Numerical Integration
right-endpoint approximation1.1 Approximating Areas
root test5.6 Ratio and Root Tests
rose7.3 Polar Coordinates
S
separable differential equation4.3 Separable Equations
separation of variables4.3 Separable Equations
Sierpinski triangle5.2 Infinite Series
sigma notation1.1 Approximating Areas
simple interest2.8 Exponential Growth and Decay
Simpson’s rule3.6 Numerical Integration
skydiver1.3 The Fundamental Theorem of Calculus
slicing method2.2 Determining Volumes by Slicing
smooth2.4 Arc Length of a Curve and Surface Area
solid of revolution2.2 Determining Volumes by Slicing
solution concentrations4.3 Separable Equations
solution curve4.2 Direction Fields and Numerical Methods
solution to a differential equation4.1 Basics of Differential Equations
space-filling curve7.3 Polar Coordinates
space-filling curves7.1 Parametric Equations
spring constant2.5 Physical Applications
standard form4.5 First-order Linear Equations, 7.5 Conic Sections
step size4.2 Direction Fields and Numerical Methods
summation notation1.1 Approximating Areas
sums and powers of integers1.1 Approximating Areas
Surface area2.4 Arc Length of a Curve and Surface Area
symmetry7.3 Polar Coordinates
symmetry principle2.6 Moments and Centers of Mass
T
Taylor polynomials6.3 Taylor and Maclaurin Series
Taylor series6.3 Taylor and Maclaurin Series
Taylor’s theorem with remainder6.3 Taylor and Maclaurin Series
telescoping series5.2 Infinite Series
term5.1 Sequences
term-by-term differentiation of a power series6.2 Properties of Power Series
term-by-term integration of a power series6.2 Properties of Power Series
theorem of Pappus for volume2.6 Moments and Centers of Mass
threshold population4.4 The Logistic Equation
total area1.2 The Definite Integral
Tour de France1.4 Integration Formulas and the Net Change Theorem
traffic accidents3.7 Improper Integrals
trapezoidal rule3.6 Numerical Integration
trigonometric integrals3.2 Trigonometric Integrals
trigonometric substitution3.3 Trigonometric Substitution
U
unbounded sequence5.1 Sequences
upper sum1.1 Approximating Areas
V
variable of integration1.2 The Definite Integral
velocity1.4 Integration Formulas and the Net Change Theorem
vertex7.5 Conic Sections
von Bertalanffy growth5.2 Infinite Series
W
washer method2.2 Determining Volumes by Slicing
wingsuits1.3 The Fundamental Theorem of Calculus
witch of Agnesi7.1 Parametric Equations
work2.5 Physical Applications
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