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Index
Cover Page Title: Calculus Know-It-ALL - Beginner to Advanced, and Everything in Between ISBN 0071549315 About the Author Contents (with page links)
Part 1 Differentiation in One Variable Part 2 Integration in One Variable Part 3 Advanced Topics
Part 1 Differentiation in One Variable Part 2 Integration in One Variable Part 3 Advanced Topics Preface Acknowledgment Calculus Know-It-ALL PART 1 Differentiation in One Variable
1 Single-Variable Functions
Mappings
Domain, range, and variables Ordered pairs Modifying a relation Three physical examples A mathematical example The inverse of a relation
Domain, range, and variables Ordered pairs Modifying a relation Three physical examples A mathematical example The inverse of a relation Linear Functions
Slope and intercept Standard form for a linear function
Slope and intercept Standard form for a linear function Nonlinear Functions
Square the input Cube the input
Square the input Cube the input “Broken” Functions
A three-part function The reciprocal function The tangent function
A three-part function The reciprocal function The tangent function Practice Exercises
Mappings
Domain, range, and variables Ordered pairs Modifying a relation Three physical examples A mathematical example The inverse of a relation
Domain, range, and variables Ordered pairs Modifying a relation Three physical examples A mathematical example The inverse of a relation Linear Functions
Slope and intercept Standard form for a linear function
Slope and intercept Standard form for a linear function Nonlinear Functions
Square the input Cube the input
Square the input Cube the input “Broken” Functions
A three-part function The reciprocal function The tangent function
A three-part function The reciprocal function The tangent function Practice Exercises 2 Limits and Continuity
Concept of the Limit
Limit of an infinite sequence Limit of a function Sum rule for two limits Multiplication-by-constant rule for a limit
Limit of an infinite sequence Limit of a function Sum rule for two limits Multiplication-by-constant rule for a limit Continuity at a Point
Right-hand limit at a point Right-hand continuity at a point Left-hand limit at a point Left-hand continuity at a point “Total” continuity at a point
Right-hand limit at a point Right-hand continuity at a point Left-hand limit at a point Left-hand continuity at a point “Total” continuity at a point Continuity of a Function
Linear functions Quadratic functions Cubic functions Polynomial functions Other continuous functions Discontinuous functions
Linear functions Quadratic functions Cubic functions Polynomial functions Other continuous functions Discontinuous functions Practice Exercises
Concept of the Limit
Limit of an infinite sequence Limit of a function Sum rule for two limits Multiplication-by-constant rule for a limit
Limit of an infinite sequence Limit of a function Sum rule for two limits Multiplication-by-constant rule for a limit Continuity at a Point
Right-hand limit at a point Right-hand continuity at a point Left-hand limit at a point Left-hand continuity at a point “Total” continuity at a point
Right-hand limit at a point Right-hand continuity at a point Left-hand limit at a point Left-hand continuity at a point “Total” continuity at a point Continuity of a Function
Linear functions Quadratic functions Cubic functions Polynomial functions Other continuous functions Discontinuous functions
Linear functions Quadratic functions Cubic functions Polynomial functions Other continuous functions Discontinuous functions Practice Exercises 3 What’s a Derivative?
Vanishing Increments
What is a tangent line? Slope between two points Converging points
What is a tangent line? Slope between two points Converging points Basic Linear Functions
Simply a constant Multiply x by a real constant
Simply a constant Multiply x by a real constant Basic Quadratic Functions
Square x and multiply by constant
Square x and multiply by constant Basic Cubic Functions
Cube x and multiply by constant
Cube x and multiply by constant Practice Exercises
Vanishing Increments
What is a tangent line? Slope between two points Converging points
What is a tangent line? Slope between two points Converging points Basic Linear Functions
Simply a constant Multiply x by a real constant
Simply a constant Multiply x by a real constant Basic Quadratic Functions
Square x and multiply by constant
Square x and multiply by constant Basic Cubic Functions
Cube x and multiply by constant
Cube x and multiply by constant Practice Exercises 4 Derivatives Don’t Always Exist
Let’s Look at the Graph
Is there a gap? Is there a jump? Is there a corner?
Is there a gap? Is there a jump? Is there a corner? When We Can Differentiate
Try to find the limits Example Another example
Try to find the limits Example Another example When We Can’t Differentiate
Example Another example
Example Another example Practice Exercises
Let’s Look at the Graph
Is there a gap? Is there a jump? Is there a corner?
Is there a gap? Is there a jump? Is there a corner? When We Can Differentiate
Try to find the limits Example Another example
Try to find the limits Example Another example When We Can’t Differentiate
Example Another example
Example Another example Practice Exercises 5 Differentiating Polynomial Functions
Power Rule
Power of a binomial Deriving the rule
Power of a binomial Deriving the rule Sum Rule
Definition and notation The rule in brief Example Another example Mathematical induction
Definition and notation The rule in brief Example Another example Mathematical induction Summing the Powers
How it looks Break it down Differentiate each term Put it back together
How it looks Break it down Differentiate each term Put it back together Practice Exercises
Power Rule
Power of a binomial Deriving the rule
Power of a binomial Deriving the rule Sum Rule
Definition and notation The rule in brief Example Another example Mathematical induction
Definition and notation The rule in brief Example Another example Mathematical induction Summing the Powers
How it looks Break it down Differentiate each term Put it back together
How it looks Break it down Differentiate each term Put it back together Practice Exercises 6 More Rules for Differentiation
Multiplication-by-Constant Rule
Definition and notation The rule in brief Example Another example
Definition and notation The rule in brief Example Another example Product Rule
Definition and notation The rule in brief Example Another example
Definition and notation The rule in brief Example Another example Reciprocal Rule
Definition and notation Caution! The rule in brief More caution! Example Another example
Definition and notation Caution! The rule in brief More caution! Example Another example Quotient Rule
Definition and notation The rule in brief Example Another example
Definition and notation The rule in brief Example Another example Chain Rule
Definition and notation The rule in brief Example Another example
Definition and notation The rule in brief Example Another example Practice Exercises
Multiplication-by-Constant Rule
Definition and notation The rule in brief Example Another example
Definition and notation The rule in brief Example Another example Product Rule
Definition and notation The rule in brief Example Another example
Definition and notation The rule in brief Example Another example Reciprocal Rule
Definition and notation Caution! The rule in brief More caution! Example Another example
Definition and notation Caution! The rule in brief More caution! Example Another example Quotient Rule
Definition and notation The rule in brief Example Another example
Definition and notation The rule in brief Example Another example Chain Rule
Definition and notation The rule in brief Example Another example
Definition and notation The rule in brief Example Another example Practice Exercises 7 A Few More Derivatives
Real-Power Rule
The old rule extended Three examples
The old rule extended Three examples Sine and Cosine Functions
What’s the sine? What’s the cosine? Derivative of the sine Derivative of the cosine Example Another example Be warned!
What’s the sine? What’s the cosine? Derivative of the sine Derivative of the cosine Example Another example Be warned! Natural Exponential Function
What’s a natural exponential? What does it look like? Offspring functions Example
What’s a natural exponential? What does it look like? Offspring functions Example Natural Logarithm Function
What’s a natural log? What does it look like? Example Where to find more derivatives Real powers in general
What’s a natural log? What does it look like? Example Where to find more derivatives Real powers in general Practice Exercises
Real-Power Rule
The old rule extended Three examples
The old rule extended Three examples Sine and Cosine Functions
What’s the sine? What’s the cosine? Derivative of the sine Derivative of the cosine Example Another example Be warned!
What’s the sine? What’s the cosine? Derivative of the sine Derivative of the cosine Example Another example Be warned! Natural Exponential Function
What’s a natural exponential? What does it look like? Offspring functions Example
What’s a natural exponential? What does it look like? Offspring functions Example Natural Logarithm Function
What’s a natural log? What does it look like? Example Where to find more derivatives Real powers in general
What’s a natural log? What does it look like? Example Where to find more derivatives Real powers in general Practice Exercises 8 Higher Derivatives
Second Derivative
An example Another example Still another example
An example Another example Still another example Third Derivative
An example Another example Still another example
An example Another example Still another example Beyond the Third Derivative
An example Another example Still another example
An example Another example Still another example Practice Exercises
Second Derivative
An example Another example Still another example
An example Another example Still another example Third Derivative
An example Another example Still another example
An example Another example Still another example Beyond the Third Derivative
An example Another example Still another example
An example Another example Still another example Practice Exercises 9 Analyzing Graphs with Derivatives
Three Common Traits
Concavity Extrema Inflection point
Concavity Extrema Inflection point Graph of a Quadratic Function
An example
An example Graph of a Cubic Function
Example
Example Graph of the Sine Function
Analyzing the curve
Analyzing the curve Practice Exercises
Three Common Traits
Concavity Extrema Inflection point
Concavity Extrema Inflection point Graph of a Quadratic Function
An example
An example Graph of a Cubic Function
Example
Example Graph of the Sine Function
Analyzing the curve
Analyzing the curve Practice Exercises 10 Review Questions and Answers
Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9
Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9
1 Single-Variable Functions
Mappings
Domain, range, and variables Ordered pairs Modifying a relation Three physical examples A mathematical example The inverse of a relation
Domain, range, and variables Ordered pairs Modifying a relation Three physical examples A mathematical example The inverse of a relation Linear Functions
Slope and intercept Standard form for a linear function
Slope and intercept Standard form for a linear function Nonlinear Functions
Square the input Cube the input
Square the input Cube the input “Broken” Functions
A three-part function The reciprocal function The tangent function
A three-part function The reciprocal function The tangent function Practice Exercises
Mappings
Domain, range, and variables Ordered pairs Modifying a relation Three physical examples A mathematical example The inverse of a relation
Domain, range, and variables Ordered pairs Modifying a relation Three physical examples A mathematical example The inverse of a relation Linear Functions
Slope and intercept Standard form for a linear function
Slope and intercept Standard form for a linear function Nonlinear Functions
Square the input Cube the input
Square the input Cube the input “Broken” Functions
A three-part function The reciprocal function The tangent function
A three-part function The reciprocal function The tangent function Practice Exercises 2 Limits and Continuity
Concept of the Limit
Limit of an infinite sequence Limit of a function Sum rule for two limits Multiplication-by-constant rule for a limit
Limit of an infinite sequence Limit of a function Sum rule for two limits Multiplication-by-constant rule for a limit Continuity at a Point
Right-hand limit at a point Right-hand continuity at a point Left-hand limit at a point Left-hand continuity at a point “Total” continuity at a point
Right-hand limit at a point Right-hand continuity at a point Left-hand limit at a point Left-hand continuity at a point “Total” continuity at a point Continuity of a Function
Linear functions Quadratic functions Cubic functions Polynomial functions Other continuous functions Discontinuous functions
Linear functions Quadratic functions Cubic functions Polynomial functions Other continuous functions Discontinuous functions Practice Exercises
Concept of the Limit
Limit of an infinite sequence Limit of a function Sum rule for two limits Multiplication-by-constant rule for a limit
Limit of an infinite sequence Limit of a function Sum rule for two limits Multiplication-by-constant rule for a limit Continuity at a Point
Right-hand limit at a point Right-hand continuity at a point Left-hand limit at a point Left-hand continuity at a point “Total” continuity at a point
Right-hand limit at a point Right-hand continuity at a point Left-hand limit at a point Left-hand continuity at a point “Total” continuity at a point Continuity of a Function
Linear functions Quadratic functions Cubic functions Polynomial functions Other continuous functions Discontinuous functions
Linear functions Quadratic functions Cubic functions Polynomial functions Other continuous functions Discontinuous functions Practice Exercises 3 What’s a Derivative?
Vanishing Increments
What is a tangent line? Slope between two points Converging points
What is a tangent line? Slope between two points Converging points Basic Linear Functions
Simply a constant Multiply x by a real constant
Simply a constant Multiply x by a real constant Basic Quadratic Functions
Square x and multiply by constant
Square x and multiply by constant Basic Cubic Functions
Cube x and multiply by constant
Cube x and multiply by constant Practice Exercises
Vanishing Increments
What is a tangent line? Slope between two points Converging points
What is a tangent line? Slope between two points Converging points Basic Linear Functions
Simply a constant Multiply x by a real constant
Simply a constant Multiply x by a real constant Basic Quadratic Functions
Square x and multiply by constant
Square x and multiply by constant Basic Cubic Functions
Cube x and multiply by constant
Cube x and multiply by constant Practice Exercises 4 Derivatives Don’t Always Exist
Let’s Look at the Graph
Is there a gap? Is there a jump? Is there a corner?
Is there a gap? Is there a jump? Is there a corner? When We Can Differentiate
Try to find the limits Example Another example
Try to find the limits Example Another example When We Can’t Differentiate
Example Another example
Example Another example Practice Exercises
Let’s Look at the Graph
Is there a gap? Is there a jump? Is there a corner?
Is there a gap? Is there a jump? Is there a corner? When We Can Differentiate
Try to find the limits Example Another example
Try to find the limits Example Another example When We Can’t Differentiate
Example Another example
Example Another example Practice Exercises 5 Differentiating Polynomial Functions
Power Rule
Power of a binomial Deriving the rule
Power of a binomial Deriving the rule Sum Rule
Definition and notation The rule in brief Example Another example Mathematical induction
Definition and notation The rule in brief Example Another example Mathematical induction Summing the Powers
How it looks Break it down Differentiate each term Put it back together
How it looks Break it down Differentiate each term Put it back together Practice Exercises
Power Rule
Power of a binomial Deriving the rule
Power of a binomial Deriving the rule Sum Rule
Definition and notation The rule in brief Example Another example Mathematical induction
Definition and notation The rule in brief Example Another example Mathematical induction Summing the Powers
How it looks Break it down Differentiate each term Put it back together
How it looks Break it down Differentiate each term Put it back together Practice Exercises 6 More Rules for Differentiation
Multiplication-by-Constant Rule
Definition and notation The rule in brief Example Another example
Definition and notation The rule in brief Example Another example Product Rule
Definition and notation The rule in brief Example Another example
Definition and notation The rule in brief Example Another example Reciprocal Rule
Definition and notation Caution! The rule in brief More caution! Example Another example
Definition and notation Caution! The rule in brief More caution! Example Another example Quotient Rule
Definition and notation The rule in brief Example Another example
Definition and notation The rule in brief Example Another example Chain Rule
Definition and notation The rule in brief Example Another example
Definition and notation The rule in brief Example Another example Practice Exercises
Multiplication-by-Constant Rule
Definition and notation The rule in brief Example Another example
Definition and notation The rule in brief Example Another example Product Rule
Definition and notation The rule in brief Example Another example
Definition and notation The rule in brief Example Another example Reciprocal Rule
Definition and notation Caution! The rule in brief More caution! Example Another example
Definition and notation Caution! The rule in brief More caution! Example Another example Quotient Rule
Definition and notation The rule in brief Example Another example
Definition and notation The rule in brief Example Another example Chain Rule
Definition and notation The rule in brief Example Another example
Definition and notation The rule in brief Example Another example Practice Exercises 7 A Few More Derivatives
Real-Power Rule
The old rule extended Three examples
The old rule extended Three examples Sine and Cosine Functions
What’s the sine? What’s the cosine? Derivative of the sine Derivative of the cosine Example Another example Be warned!
What’s the sine? What’s the cosine? Derivative of the sine Derivative of the cosine Example Another example Be warned! Natural Exponential Function
What’s a natural exponential? What does it look like? Offspring functions Example
What’s a natural exponential? What does it look like? Offspring functions Example Natural Logarithm Function
What’s a natural log? What does it look like? Example Where to find more derivatives Real powers in general
What’s a natural log? What does it look like? Example Where to find more derivatives Real powers in general Practice Exercises
Real-Power Rule
The old rule extended Three examples
The old rule extended Three examples Sine and Cosine Functions
What’s the sine? What’s the cosine? Derivative of the sine Derivative of the cosine Example Another example Be warned!
What’s the sine? What’s the cosine? Derivative of the sine Derivative of the cosine Example Another example Be warned! Natural Exponential Function
What’s a natural exponential? What does it look like? Offspring functions Example
What’s a natural exponential? What does it look like? Offspring functions Example Natural Logarithm Function
What’s a natural log? What does it look like? Example Where to find more derivatives Real powers in general
What’s a natural log? What does it look like? Example Where to find more derivatives Real powers in general Practice Exercises 8 Higher Derivatives
Second Derivative
An example Another example Still another example
An example Another example Still another example Third Derivative
An example Another example Still another example
An example Another example Still another example Beyond the Third Derivative
An example Another example Still another example
An example Another example Still another example Practice Exercises
Second Derivative
An example Another example Still another example
An example Another example Still another example Third Derivative
An example Another example Still another example
An example Another example Still another example Beyond the Third Derivative
An example Another example Still another example
An example Another example Still another example Practice Exercises 9 Analyzing Graphs with Derivatives
Three Common Traits
Concavity Extrema Inflection point
Concavity Extrema Inflection point Graph of a Quadratic Function
An example
An example Graph of a Cubic Function
Example
Example Graph of the Sine Function
Analyzing the curve
Analyzing the curve Practice Exercises
Three Common Traits
Concavity Extrema Inflection point
Concavity Extrema Inflection point Graph of a Quadratic Function
An example
An example Graph of a Cubic Function
Example
Example Graph of the Sine Function
Analyzing the curve
Analyzing the curve Practice Exercises 10 Review Questions and Answers
Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9
Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 PART 2 Integration in One Variable
11 What’s an Integral?
Summation Notation
Specify the series Tag the terms The big sigma A more sophisticated example
Specify the series Tag the terms The big sigma A more sophisticated example Area Defined by a Curve
Defining the region Approximating the area The shrinking increment The Riemann magic The integral notation
Defining the region Approximating the area The shrinking increment The Riemann magic The integral notation Three Applications
Displacement vs. speed Average value Normal distribution
Displacement vs. speed Average value Normal distribution Practice Exercises
Summation Notation
Specify the series Tag the terms The big sigma A more sophisticated example
Specify the series Tag the terms The big sigma A more sophisticated example Area Defined by a Curve
Defining the region Approximating the area The shrinking increment The Riemann magic The integral notation
Defining the region Approximating the area The shrinking increment The Riemann magic The integral notation Three Applications
Displacement vs. speed Average value Normal distribution
Displacement vs. speed Average value Normal distribution Practice Exercises 12 Derivatives in Reverse
Concept of the Antiderivative
The notation Antiderivatives of the zero function
The notation Antiderivatives of the zero function Some Simple Antiderivatives
Antiderivatives of nonzero constant functions Antiderivatives of basic linear functions Antiderivatives of basic quadratic functions Antiderivatives of basic nth-degree functions Real-number exponents
Antiderivatives of nonzero constant functions Antiderivatives of basic linear functions Antiderivatives of basic quadratic functions Antiderivatives of basic nth-degree functions Real-number exponents Indefinite Integral
The notation An example “Pulling out” a constant “Pulling out” the negative Sum of indefinite integrals
The notation An example “Pulling out” a constant “Pulling out” the negative Sum of indefinite integrals Definite Integral
The Fundamental Theorem of Calculus An example Another example
The Fundamental Theorem of Calculus An example Another example Practice Exercises
Concept of the Antiderivative
The notation Antiderivatives of the zero function
The notation Antiderivatives of the zero function Some Simple Antiderivatives
Antiderivatives of nonzero constant functions Antiderivatives of basic linear functions Antiderivatives of basic quadratic functions Antiderivatives of basic nth-degree functions Real-number exponents
Antiderivatives of nonzero constant functions Antiderivatives of basic linear functions Antiderivatives of basic quadratic functions Antiderivatives of basic nth-degree functions Real-number exponents Indefinite Integral
The notation An example “Pulling out” a constant “Pulling out” the negative Sum of indefinite integrals
The notation An example “Pulling out” a constant “Pulling out” the negative Sum of indefinite integrals Definite Integral
The Fundamental Theorem of Calculus An example Another example
The Fundamental Theorem of Calculus An example Another example Practice Exercises 13 Three Rules for Integration
Reversal Rule
The rule in brief Example Another example
The rule in brief Example Another example Split-Interval Rule
The rule in brief Example Another example
The rule in brief Example Another example Substitution Rule
The rule in brief Example
The rule in brief Example Practice Exercises
Reversal Rule
The rule in brief Example Another example
The rule in brief Example Another example Split-Interval Rule
The rule in brief Example Another example
The rule in brief Example Another example Substitution Rule
The rule in brief Example
The rule in brief Example Practice Exercises 14 Improper Integrals
Variable Bounds
Adjusting the upper bound Adjusting the lower bound “Runaway” bounds
Adjusting the upper bound Adjusting the lower bound “Runaway” bounds Singularity in the Interval
How to do it wrong How to do it correctly: the left-hand side How to do it correctly: the right-hand side
How to do it wrong How to do it correctly: the left-hand side How to do it correctly: the right-hand side Infinite Intervals
Example Another example
Example Another example Practice Exercises
Variable Bounds
Adjusting the upper bound Adjusting the lower bound “Runaway” bounds
Adjusting the upper bound Adjusting the lower bound “Runaway” bounds Singularity in the Interval
How to do it wrong How to do it correctly: the left-hand side How to do it correctly: the right-hand side
How to do it wrong How to do it correctly: the left-hand side How to do it correctly: the right-hand side Infinite Intervals
Example Another example
Example Another example Practice Exercises 15 Integrating Polynomial Functions
Three Rules Revisited
The old rules The new rules
The old rules The new rules Indefinite-Integral Situations
Example Another example
Example Another example Definite-Integral Situations
Example Another example
Example Another example Practice Exercises
Three Rules Revisited
The old rules The new rules
The old rules The new rules Indefinite-Integral Situations
Example Another example
Example Another example Definite-Integral Situations
Example Another example
Example Another example Practice Exercises 16 Areas between Graphs
Line and Curve
Solve the system Work out the geometry
Solve the system Work out the geometry Two Curves
Two “mirrored” parabolas Integrate the difference function
Two “mirrored” parabolas Integrate the difference function Singular Curves
An infinitely tall zone Determine the interval Integrate the difference function
An infinitely tall zone Determine the interval Integrate the difference function Practice Exercises
Line and Curve
Solve the system Work out the geometry
Solve the system Work out the geometry Two Curves
Two “mirrored” parabolas Integrate the difference function
Two “mirrored” parabolas Integrate the difference function Singular Curves
An infinitely tall zone Determine the interval Integrate the difference function
An infinitely tall zone Determine the interval Integrate the difference function Practice Exercises 17 A Few More Integrals
Sine and Cosine Functions
Indefinite integrals of cosine and sine Example Another example
Indefinite integrals of cosine and sine Example Another example Natural Exponential Function
Indefinite integrals of basic exponential functions Example Another example
Indefinite integrals of basic exponential functions Example Another example Reciprocal Function
Indefinite integral of basic reciprocal function Example Another example Where to find more integrals
Indefinite integral of basic reciprocal function Example Another example Where to find more integrals Practice Exercises
Sine and Cosine Functions
Indefinite integrals of cosine and sine Example Another example
Indefinite integrals of cosine and sine Example Another example Natural Exponential Function
Indefinite integrals of basic exponential functions Example Another example
Indefinite integrals of basic exponential functions Example Another example Reciprocal Function
Indefinite integral of basic reciprocal function Example Another example Where to find more integrals
Indefinite integral of basic reciprocal function Example Another example Where to find more integrals Practice Exercises 18 How Long Is the Arc?
A Chorus of Chords
Breaking up the arc The law of the mean Finding the true arc length
Breaking up the arc The law of the mean Finding the true arc length A Monomial Curve
Arc-in-a-box method Setting up the integral Working out the integral
Arc-in-a-box method Setting up the integral Working out the integral A More Exotic Curve
Setting up the integral Working out the value
Setting up the integral Working out the value Practice Exercises
A Chorus of Chords
Breaking up the arc The law of the mean Finding the true arc length
Breaking up the arc The law of the mean Finding the true arc length A Monomial Curve
Arc-in-a-box method Setting up the integral Working out the integral
Arc-in-a-box method Setting up the integral Working out the integral A More Exotic Curve
Setting up the integral Working out the value
Setting up the integral Working out the value Practice Exercises 19 Special Integration Tricks
Principle of Linearity
The old rules The new rule in brief Example Don’t be fooled!
The old rules The new rule in brief Example Don’t be fooled! Integration by Parts
An old idea revisited Variations on a theme Example
An old idea revisited Variations on a theme Example Partial Fractions
A helpful formula A preliminary example “Reverse engineering” A working example
A helpful formula A preliminary example “Reverse engineering” A working example Practice Exercises
Principle of Linearity
The old rules The new rule in brief Example Don’t be fooled!
The old rules The new rule in brief Example Don’t be fooled! Integration by Parts
An old idea revisited Variations on a theme Example
An old idea revisited Variations on a theme Example Partial Fractions
A helpful formula A preliminary example “Reverse engineering” A working example
A helpful formula A preliminary example “Reverse engineering” A working example Practice Exercises 20 Review Questions and Answers
Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19
Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19
11 What’s an Integral?
Summation Notation
Specify the series Tag the terms The big sigma A more sophisticated example
Specify the series Tag the terms The big sigma A more sophisticated example Area Defined by a Curve
Defining the region Approximating the area The shrinking increment The Riemann magic The integral notation
Defining the region Approximating the area The shrinking increment The Riemann magic The integral notation Three Applications
Displacement vs. speed Average value Normal distribution
Displacement vs. speed Average value Normal distribution Practice Exercises
Summation Notation
Specify the series Tag the terms The big sigma A more sophisticated example
Specify the series Tag the terms The big sigma A more sophisticated example Area Defined by a Curve
Defining the region Approximating the area The shrinking increment The Riemann magic The integral notation
Defining the region Approximating the area The shrinking increment The Riemann magic The integral notation Three Applications
Displacement vs. speed Average value Normal distribution
Displacement vs. speed Average value Normal distribution Practice Exercises 12 Derivatives in Reverse
Concept of the Antiderivative
The notation Antiderivatives of the zero function
The notation Antiderivatives of the zero function Some Simple Antiderivatives
Antiderivatives of nonzero constant functions Antiderivatives of basic linear functions Antiderivatives of basic quadratic functions Antiderivatives of basic nth-degree functions Real-number exponents
Antiderivatives of nonzero constant functions Antiderivatives of basic linear functions Antiderivatives of basic quadratic functions Antiderivatives of basic nth-degree functions Real-number exponents Indefinite Integral
The notation An example “Pulling out” a constant “Pulling out” the negative Sum of indefinite integrals
The notation An example “Pulling out” a constant “Pulling out” the negative Sum of indefinite integrals Definite Integral
The Fundamental Theorem of Calculus An example Another example
The Fundamental Theorem of Calculus An example Another example Practice Exercises
Concept of the Antiderivative
The notation Antiderivatives of the zero function
The notation Antiderivatives of the zero function Some Simple Antiderivatives
Antiderivatives of nonzero constant functions Antiderivatives of basic linear functions Antiderivatives of basic quadratic functions Antiderivatives of basic nth-degree functions Real-number exponents
Antiderivatives of nonzero constant functions Antiderivatives of basic linear functions Antiderivatives of basic quadratic functions Antiderivatives of basic nth-degree functions Real-number exponents Indefinite Integral
The notation An example “Pulling out” a constant “Pulling out” the negative Sum of indefinite integrals
The notation An example “Pulling out” a constant “Pulling out” the negative Sum of indefinite integrals Definite Integral
The Fundamental Theorem of Calculus An example Another example
The Fundamental Theorem of Calculus An example Another example Practice Exercises 13 Three Rules for Integration
Reversal Rule
The rule in brief Example Another example
The rule in brief Example Another example Split-Interval Rule
The rule in brief Example Another example
The rule in brief Example Another example Substitution Rule
The rule in brief Example
The rule in brief Example Practice Exercises
Reversal Rule
The rule in brief Example Another example
The rule in brief Example Another example Split-Interval Rule
The rule in brief Example Another example
The rule in brief Example Another example Substitution Rule
The rule in brief Example
The rule in brief Example Practice Exercises 14 Improper Integrals
Variable Bounds
Adjusting the upper bound Adjusting the lower bound “Runaway” bounds
Adjusting the upper bound Adjusting the lower bound “Runaway” bounds Singularity in the Interval
How to do it wrong How to do it correctly: the left-hand side How to do it correctly: the right-hand side
How to do it wrong How to do it correctly: the left-hand side How to do it correctly: the right-hand side Infinite Intervals
Example Another example
Example Another example Practice Exercises
Variable Bounds
Adjusting the upper bound Adjusting the lower bound “Runaway” bounds
Adjusting the upper bound Adjusting the lower bound “Runaway” bounds Singularity in the Interval
How to do it wrong How to do it correctly: the left-hand side How to do it correctly: the right-hand side
How to do it wrong How to do it correctly: the left-hand side How to do it correctly: the right-hand side Infinite Intervals
Example Another example
Example Another example Practice Exercises 15 Integrating Polynomial Functions
Three Rules Revisited
The old rules The new rules
The old rules The new rules Indefinite-Integral Situations
Example Another example
Example Another example Definite-Integral Situations
Example Another example
Example Another example Practice Exercises
Three Rules Revisited
The old rules The new rules
The old rules The new rules Indefinite-Integral Situations
Example Another example
Example Another example Definite-Integral Situations
Example Another example
Example Another example Practice Exercises 16 Areas between Graphs
Line and Curve
Solve the system Work out the geometry
Solve the system Work out the geometry Two Curves
Two “mirrored” parabolas Integrate the difference function
Two “mirrored” parabolas Integrate the difference function Singular Curves
An infinitely tall zone Determine the interval Integrate the difference function
An infinitely tall zone Determine the interval Integrate the difference function Practice Exercises
Line and Curve
Solve the system Work out the geometry
Solve the system Work out the geometry Two Curves
Two “mirrored” parabolas Integrate the difference function
Two “mirrored” parabolas Integrate the difference function Singular Curves
An infinitely tall zone Determine the interval Integrate the difference function
An infinitely tall zone Determine the interval Integrate the difference function Practice Exercises 17 A Few More Integrals
Sine and Cosine Functions
Indefinite integrals of cosine and sine Example Another example
Indefinite integrals of cosine and sine Example Another example Natural Exponential Function
Indefinite integrals of basic exponential functions Example Another example
Indefinite integrals of basic exponential functions Example Another example Reciprocal Function
Indefinite integral of basic reciprocal function Example Another example Where to find more integrals
Indefinite integral of basic reciprocal function Example Another example Where to find more integrals Practice Exercises
Sine and Cosine Functions
Indefinite integrals of cosine and sine Example Another example
Indefinite integrals of cosine and sine Example Another example Natural Exponential Function
Indefinite integrals of basic exponential functions Example Another example
Indefinite integrals of basic exponential functions Example Another example Reciprocal Function
Indefinite integral of basic reciprocal function Example Another example Where to find more integrals
Indefinite integral of basic reciprocal function Example Another example Where to find more integrals Practice Exercises 18 How Long Is the Arc?
A Chorus of Chords
Breaking up the arc The law of the mean Finding the true arc length
Breaking up the arc The law of the mean Finding the true arc length A Monomial Curve
Arc-in-a-box method Setting up the integral Working out the integral
Arc-in-a-box method Setting up the integral Working out the integral A More Exotic Curve
Setting up the integral Working out the value
Setting up the integral Working out the value Practice Exercises
A Chorus of Chords
Breaking up the arc The law of the mean Finding the true arc length
Breaking up the arc The law of the mean Finding the true arc length A Monomial Curve
Arc-in-a-box method Setting up the integral Working out the integral
Arc-in-a-box method Setting up the integral Working out the integral A More Exotic Curve
Setting up the integral Working out the value
Setting up the integral Working out the value Practice Exercises 19 Special Integration Tricks
Principle of Linearity
The old rules The new rule in brief Example Don’t be fooled!
The old rules The new rule in brief Example Don’t be fooled! Integration by Parts
An old idea revisited Variations on a theme Example
An old idea revisited Variations on a theme Example Partial Fractions
A helpful formula A preliminary example “Reverse engineering” A working example
A helpful formula A preliminary example “Reverse engineering” A working example Practice Exercises
Principle of Linearity
The old rules The new rule in brief Example Don’t be fooled!
The old rules The new rule in brief Example Don’t be fooled! Integration by Parts
An old idea revisited Variations on a theme Example
An old idea revisited Variations on a theme Example Partial Fractions
A helpful formula A preliminary example “Reverse engineering” A working example
A helpful formula A preliminary example “Reverse engineering” A working example Practice Exercises 20 Review Questions and Answers
Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19
Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 PART 3 Advanced Topics
21 Differentiating Inverse Functions
A General Formula
What is an inverse function? Differentiating “through the back door” Example: “front door” Same example: “back door”
What is an inverse function? Differentiating “through the back door” Example: “front door” Same example: “back door” Derivative of the Arcsine
Restricting the domain Getting the formula
Restricting the domain Getting the formula Derivative of the Arccosine
Restricting the domain Getting the formula
Restricting the domain Getting the formula Practice Exercises
A General Formula
What is an inverse function? Differentiating “through the back door” Example: “front door” Same example: “back door”
What is an inverse function? Differentiating “through the back door” Example: “front door” Same example: “back door” Derivative of the Arcsine
Restricting the domain Getting the formula
Restricting the domain Getting the formula Derivative of the Arccosine
Restricting the domain Getting the formula
Restricting the domain Getting the formula Practice Exercises 22 Implicit Differentiation
Two-Way Relations
How shall we write it ? Equations of circles Equations of ellipses Equations of hyperbolas
How shall we write it ? Equations of circles Equations of ellipses Equations of hyperbolas Two-Way Derivatives
Example: unit circle Example: ellipse Example: hyperbola
Example: unit circle Example: ellipse Example: hyperbola Practice Exercises
Two-Way Relations
How shall we write it ? Equations of circles Equations of ellipses Equations of hyperbolas
How shall we write it ? Equations of circles Equations of ellipses Equations of hyperbolas Two-Way Derivatives
Example: unit circle Example: ellipse Example: hyperbola
Example: unit circle Example: ellipse Example: hyperbola Practice Exercises 23 The L’Hopital Principles
Expressions That Tend Toward 0/0
How it works Example Applying the rule twice An important restriction
How it works Example Applying the rule twice An important restriction Expressions That Tend Toward ±∞/±∞
How it works Example Another variant of the rule Another example
How it works Example Another variant of the rule Another example Other Indeterminate Limits
Expressions That Tend Toward 0·(+∞) or 0·(-∞) Expressions That Tend Toward +∞ - (+∞) Expressions tending toward +∞·0, -∞·0, -∞+(+∞), +∞+(-∞) or-∞(-∞)
Expressions That Tend Toward 0·(+∞) or 0·(-∞) Expressions That Tend Toward +∞ - (+∞) Expressions tending toward +∞·0, -∞·0, -∞+(+∞), +∞+(-∞) or-∞(-∞) Practice Exercises
Expressions That Tend Toward 0/0
How it works Example Applying the rule twice An important restriction
How it works Example Applying the rule twice An important restriction Expressions That Tend Toward ±∞/±∞
How it works Example Another variant of the rule Another example
How it works Example Another variant of the rule Another example Other Indeterminate Limits
Expressions That Tend Toward 0·(+∞) or 0·(-∞) Expressions That Tend Toward +∞ - (+∞) Expressions tending toward +∞·0, -∞·0, -∞+(+∞), +∞+(-∞) or-∞(-∞)
Expressions That Tend Toward 0·(+∞) or 0·(-∞) Expressions That Tend Toward +∞ - (+∞) Expressions tending toward +∞·0, -∞·0, -∞+(+∞), +∞+(-∞) or-∞(-∞) Practice Exercises 24 Partial Derivatives
Multi-Variable Functions
Two inputs, one output A pure-mathematics example The vertical-line test Three inputs, one output What about time?
Two inputs, one output A pure-mathematics example The vertical-line test Three inputs, one output What about time? Two Independent Variables
“Slope” of a surface at a point Derivative with respect to x Derivative with respect to y Example Another example
“Slope” of a surface at a point Derivative with respect to x Derivative with respect to y Example Another example Three Independent Variables
Derivatives with respect to x,y, and z Example Another example
Derivatives with respect to x,y, and z Example Another example Practice Exercises
Multi-Variable Functions
Two inputs, one output A pure-mathematics example The vertical-line test Three inputs, one output What about time?
Two inputs, one output A pure-mathematics example The vertical-line test Three inputs, one output What about time? Two Independent Variables
“Slope” of a surface at a point Derivative with respect to x Derivative with respect to y Example Another example
“Slope” of a surface at a point Derivative with respect to x Derivative with respect to y Example Another example Three Independent Variables
Derivatives with respect to x,y, and z Example Another example
Derivatives with respect to x,y, and z Example Another example Practice Exercises 25 Second Partial Derivatives
Two Variables, Second Partials
Second partials relative to x or y Example Another example
Second partials relative to x or y Example Another example Two Variables, Mixed Partials
Differentiating with respect to x and then y Differentiating with respect to y and then x Example Another example A theorem
Differentiating with respect to x and then y Differentiating with respect to y and then x Example Another example A theorem Three Variables, Second Partials
Second partials with respect to x,y, or z Example Another example
Second partials with respect to x,y, or z Example Another example Three Variables, Mixed Partials
Six ways to mix Example
Six ways to mix Example Practice Exercises
Two Variables, Second Partials
Second partials relative to x or y Example Another example
Second partials relative to x or y Example Another example Two Variables, Mixed Partials
Differentiating with respect to x and then y Differentiating with respect to y and then x Example Another example A theorem
Differentiating with respect to x and then y Differentiating with respect to y and then x Example Another example A theorem Three Variables, Second Partials
Second partials with respect to x,y, or z Example Another example
Second partials with respect to x,y, or z Example Another example Three Variables, Mixed Partials
Six ways to mix Example
Six ways to mix Example Practice Exercises 26 Surface-Area and Volume Integrals
A Cylinder
Circumference vs. displacement Lateral-surface area integral Cross-sectional area vs. displacement Volume integral
Circumference vs. displacement Lateral-surface area integral Cross-sectional area vs. displacement Volume integral A Cone
Circumference vs. displacement Slant-surface area integral Cross-sectional area vs. displacement Volume integral
Circumference vs. displacement Slant-surface area integral Cross-sectional area vs. displacement Volume integral A Sphere
Circumference vs. arc displacement Surface-area integral Cross-sectional area vs. displacement Volume integral
Circumference vs. arc displacement Surface-area integral Cross-sectional area vs. displacement Volume integral Practice Exercises
A Cylinder
Circumference vs. displacement Lateral-surface area integral Cross-sectional area vs. displacement Volume integral
Circumference vs. displacement Lateral-surface area integral Cross-sectional area vs. displacement Volume integral A Cone
Circumference vs. displacement Slant-surface area integral Cross-sectional area vs. displacement Volume integral
Circumference vs. displacement Slant-surface area integral Cross-sectional area vs. displacement Volume integral A Sphere
Circumference vs. arc displacement Surface-area integral Cross-sectional area vs. displacement Volume integral
Circumference vs. arc displacement Surface-area integral Cross-sectional area vs. displacement Volume integral Practice Exercises 27 Repeated, Double, and Iterated Integrals
Repeated Integrals in One Variable
Multiple definite integrals Example Another example
Multiple definite integrals Example Another example Double Integrals in Two Variables
Prisms and slabs Slabs parallel to the xz-plane Slabs parallel to the yz-plane
Prisms and slabs Slabs parallel to the xz-plane Slabs parallel to the yz-plane Iterated Integrals in Two Variables
Example Another example
Example Another example Practice Exercises
Repeated Integrals in One Variable
Multiple definite integrals Example Another example
Multiple definite integrals Example Another example Double Integrals in Two Variables
Prisms and slabs Slabs parallel to the xz-plane Slabs parallel to the yz-plane
Prisms and slabs Slabs parallel to the xz-plane Slabs parallel to the yz-plane Iterated Integrals in Two Variables
Example Another example
Example Another example Practice Exercises 28 More Volume Integrals
Slicing and Integrating
First, we slice Next, we integrate Finally, we integrate again
First, we slice Next, we integrate Finally, we integrate again Base Bounded by Curve and x Axis
The proper structure of Cartesian xyz-space Flat, level surface Flat, sloping surface Warped surface
The proper structure of Cartesian xyz-space Flat, level surface Flat, sloping surface Warped surface Base Bounded by Curve and Line
Flat, level surface Flat, sloping surface Warped surface
Flat, level surface Flat, sloping surface Warped surface Base Bounded by Two Curves
Flat, level surface Flat, sloping surface Warped surface
Flat, level surface Flat, sloping surface Warped surface Practice Exercises
Slicing and Integrating
First, we slice Next, we integrate Finally, we integrate again
First, we slice Next, we integrate Finally, we integrate again Base Bounded by Curve and x Axis
The proper structure of Cartesian xyz-space Flat, level surface Flat, sloping surface Warped surface
The proper structure of Cartesian xyz-space Flat, level surface Flat, sloping surface Warped surface Base Bounded by Curve and Line
Flat, level surface Flat, sloping surface Warped surface
Flat, level surface Flat, sloping surface Warped surface Base Bounded by Two Curves
Flat, level surface Flat, sloping surface Warped surface
Flat, level surface Flat, sloping surface Warped surface Practice Exercises 29 What’s a Differential Equation?
Elementary First-Order ODEs
How to recognize one Example 1 Example 2 Example 3
How to recognize one Example 1 Example 2 Example 3 Elementary Second-Order ODEs
How to recognize one Example 4 Example 5 Example 6 For further study
How to recognize one Example 4 Example 5 Example 6 For further study Practice Exercises
Elementary First-Order ODEs
How to recognize one Example 1 Example 2 Example 3
How to recognize one Example 1 Example 2 Example 3 Elementary Second-Order ODEs
How to recognize one Example 4 Example 5 Example 6 For further study
How to recognize one Example 4 Example 5 Example 6 For further study Practice Exercises 30 Review Questions and Answers
Chapter 21 Chapter 22 Chapter 23 Chapter 24 Chapter 25 Chapter 26 Chapter 27 Chapter 28 Chapter 29
Chapter 21 Chapter 22 Chapter 23 Chapter 24 Chapter 25 Chapter 26 Chapter 27 Chapter 28 Chapter 29
21 Differentiating Inverse Functions
A General Formula
What is an inverse function? Differentiating “through the back door” Example: “front door” Same example: “back door”
What is an inverse function? Differentiating “through the back door” Example: “front door” Same example: “back door” Derivative of the Arcsine
Restricting the domain Getting the formula
Restricting the domain Getting the formula Derivative of the Arccosine
Restricting the domain Getting the formula
Restricting the domain Getting the formula Practice Exercises
A General Formula
What is an inverse function? Differentiating “through the back door” Example: “front door” Same example: “back door”
What is an inverse function? Differentiating “through the back door” Example: “front door” Same example: “back door” Derivative of the Arcsine
Restricting the domain Getting the formula
Restricting the domain Getting the formula Derivative of the Arccosine
Restricting the domain Getting the formula
Restricting the domain Getting the formula Practice Exercises 22 Implicit Differentiation
Two-Way Relations
How shall we write it ? Equations of circles Equations of ellipses Equations of hyperbolas
How shall we write it ? Equations of circles Equations of ellipses Equations of hyperbolas Two-Way Derivatives
Example: unit circle Example: ellipse Example: hyperbola
Example: unit circle Example: ellipse Example: hyperbola Practice Exercises
Two-Way Relations
How shall we write it ? Equations of circles Equations of ellipses Equations of hyperbolas
How shall we write it ? Equations of circles Equations of ellipses Equations of hyperbolas Two-Way Derivatives
Example: unit circle Example: ellipse Example: hyperbola
Example: unit circle Example: ellipse Example: hyperbola Practice Exercises 23 The L’Hopital Principles
Expressions That Tend Toward 0/0
How it works Example Applying the rule twice An important restriction
How it works Example Applying the rule twice An important restriction Expressions That Tend Toward ±∞/±∞
How it works Example Another variant of the rule Another example
How it works Example Another variant of the rule Another example Other Indeterminate Limits
Expressions That Tend Toward 0·(+∞) or 0·(-∞) Expressions That Tend Toward +∞ - (+∞) Expressions tending toward +∞·0, -∞·0, -∞+(+∞), +∞+(-∞) or-∞(-∞)
Expressions That Tend Toward 0·(+∞) or 0·(-∞) Expressions That Tend Toward +∞ - (+∞) Expressions tending toward +∞·0, -∞·0, -∞+(+∞), +∞+(-∞) or-∞(-∞) Practice Exercises
Expressions That Tend Toward 0/0
How it works Example Applying the rule twice An important restriction
How it works Example Applying the rule twice An important restriction Expressions That Tend Toward ±∞/±∞
How it works Example Another variant of the rule Another example
How it works Example Another variant of the rule Another example Other Indeterminate Limits
Expressions That Tend Toward 0·(+∞) or 0·(-∞) Expressions That Tend Toward +∞ - (+∞) Expressions tending toward +∞·0, -∞·0, -∞+(+∞), +∞+(-∞) or-∞(-∞)
Expressions That Tend Toward 0·(+∞) or 0·(-∞) Expressions That Tend Toward +∞ - (+∞) Expressions tending toward +∞·0, -∞·0, -∞+(+∞), +∞+(-∞) or-∞(-∞) Practice Exercises 24 Partial Derivatives
Multi-Variable Functions
Two inputs, one output A pure-mathematics example The vertical-line test Three inputs, one output What about time?
Two inputs, one output A pure-mathematics example The vertical-line test Three inputs, one output What about time? Two Independent Variables
“Slope” of a surface at a point Derivative with respect to x Derivative with respect to y Example Another example
“Slope” of a surface at a point Derivative with respect to x Derivative with respect to y Example Another example Three Independent Variables
Derivatives with respect to x,y, and z Example Another example
Derivatives with respect to x,y, and z Example Another example Practice Exercises
Multi-Variable Functions
Two inputs, one output A pure-mathematics example The vertical-line test Three inputs, one output What about time?
Two inputs, one output A pure-mathematics example The vertical-line test Three inputs, one output What about time? Two Independent Variables
“Slope” of a surface at a point Derivative with respect to x Derivative with respect to y Example Another example
“Slope” of a surface at a point Derivative with respect to x Derivative with respect to y Example Another example Three Independent Variables
Derivatives with respect to x,y, and z Example Another example
Derivatives with respect to x,y, and z Example Another example Practice Exercises 25 Second Partial Derivatives
Two Variables, Second Partials
Second partials relative to x or y Example Another example
Second partials relative to x or y Example Another example Two Variables, Mixed Partials
Differentiating with respect to x and then y Differentiating with respect to y and then x Example Another example A theorem
Differentiating with respect to x and then y Differentiating with respect to y and then x Example Another example A theorem Three Variables, Second Partials
Second partials with respect to x,y, or z Example Another example
Second partials with respect to x,y, or z Example Another example Three Variables, Mixed Partials
Six ways to mix Example
Six ways to mix Example Practice Exercises
Two Variables, Second Partials
Second partials relative to x or y Example Another example
Second partials relative to x or y Example Another example Two Variables, Mixed Partials
Differentiating with respect to x and then y Differentiating with respect to y and then x Example Another example A theorem
Differentiating with respect to x and then y Differentiating with respect to y and then x Example Another example A theorem Three Variables, Second Partials
Second partials with respect to x,y, or z Example Another example
Second partials with respect to x,y, or z Example Another example Three Variables, Mixed Partials
Six ways to mix Example
Six ways to mix Example Practice Exercises 26 Surface-Area and Volume Integrals
A Cylinder
Circumference vs. displacement Lateral-surface area integral Cross-sectional area vs. displacement Volume integral
Circumference vs. displacement Lateral-surface area integral Cross-sectional area vs. displacement Volume integral A Cone
Circumference vs. displacement Slant-surface area integral Cross-sectional area vs. displacement Volume integral
Circumference vs. displacement Slant-surface area integral Cross-sectional area vs. displacement Volume integral A Sphere
Circumference vs. arc displacement Surface-area integral Cross-sectional area vs. displacement Volume integral
Circumference vs. arc displacement Surface-area integral Cross-sectional area vs. displacement Volume integral Practice Exercises
A Cylinder
Circumference vs. displacement Lateral-surface area integral Cross-sectional area vs. displacement Volume integral
Circumference vs. displacement Lateral-surface area integral Cross-sectional area vs. displacement Volume integral A Cone
Circumference vs. displacement Slant-surface area integral Cross-sectional area vs. displacement Volume integral
Circumference vs. displacement Slant-surface area integral Cross-sectional area vs. displacement Volume integral A Sphere
Circumference vs. arc displacement Surface-area integral Cross-sectional area vs. displacement Volume integral
Circumference vs. arc displacement Surface-area integral Cross-sectional area vs. displacement Volume integral Practice Exercises 27 Repeated, Double, and Iterated Integrals
Repeated Integrals in One Variable
Multiple definite integrals Example Another example
Multiple definite integrals Example Another example Double Integrals in Two Variables
Prisms and slabs Slabs parallel to the xz-plane Slabs parallel to the yz-plane
Prisms and slabs Slabs parallel to the xz-plane Slabs parallel to the yz-plane Iterated Integrals in Two Variables
Example Another example
Example Another example Practice Exercises
Repeated Integrals in One Variable
Multiple definite integrals Example Another example
Multiple definite integrals Example Another example Double Integrals in Two Variables
Prisms and slabs Slabs parallel to the xz-plane Slabs parallel to the yz-plane
Prisms and slabs Slabs parallel to the xz-plane Slabs parallel to the yz-plane Iterated Integrals in Two Variables
Example Another example
Example Another example Practice Exercises 28 More Volume Integrals
Slicing and Integrating
First, we slice Next, we integrate Finally, we integrate again
First, we slice Next, we integrate Finally, we integrate again Base Bounded by Curve and x Axis
The proper structure of Cartesian xyz-space Flat, level surface Flat, sloping surface Warped surface
The proper structure of Cartesian xyz-space Flat, level surface Flat, sloping surface Warped surface Base Bounded by Curve and Line
Flat, level surface Flat, sloping surface Warped surface
Flat, level surface Flat, sloping surface Warped surface Base Bounded by Two Curves
Flat, level surface Flat, sloping surface Warped surface
Flat, level surface Flat, sloping surface Warped surface Practice Exercises
Slicing and Integrating
First, we slice Next, we integrate Finally, we integrate again
First, we slice Next, we integrate Finally, we integrate again Base Bounded by Curve and x Axis
The proper structure of Cartesian xyz-space Flat, level surface Flat, sloping surface Warped surface
The proper structure of Cartesian xyz-space Flat, level surface Flat, sloping surface Warped surface Base Bounded by Curve and Line
Flat, level surface Flat, sloping surface Warped surface
Flat, level surface Flat, sloping surface Warped surface Base Bounded by Two Curves
Flat, level surface Flat, sloping surface Warped surface
Flat, level surface Flat, sloping surface Warped surface Practice Exercises 29 What’s a Differential Equation?
Elementary First-Order ODEs
How to recognize one Example 1 Example 2 Example 3
How to recognize one Example 1 Example 2 Example 3 Elementary Second-Order ODEs
How to recognize one Example 4 Example 5 Example 6 For further study
How to recognize one Example 4 Example 5 Example 6 For further study Practice Exercises
Elementary First-Order ODEs
How to recognize one Example 1 Example 2 Example 3
How to recognize one Example 1 Example 2 Example 3 Elementary Second-Order ODEs
How to recognize one Example 4 Example 5 Example 6 For further study
How to recognize one Example 4 Example 5 Example 6 For further study Practice Exercises 30 Review Questions and Answers
Chapter 21 Chapter 22 Chapter 23 Chapter 24 Chapter 25 Chapter 26 Chapter 27 Chapter 28 Chapter 29
Chapter 21 Chapter 22 Chapter 23 Chapter 24 Chapter 25 Chapter 26 Chapter 27 Chapter 28 Chapter 29 Final Exam APPENDICES
A Worked-Out Solutions to Exercises: Chapters 1 to 9
Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9
Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 B Worked-Out Solutions to Exercises: Chapters 11 to 19
Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19
Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 C Worked-Out Solutions to Exercises: Chapters 21 to 29
Chapter 21 Chapter 22 Chapter 23 Chapter 24 Chapter 25 Chapter 26 Chapter 27 Chapter 28 Chapter 29
Chapter 21 Chapter 22 Chapter 23 Chapter 24 Chapter 25 Chapter 26 Chapter 27 Chapter 28 Chapter 29 D Answers to Final Exam Questions E Special Characters in Order of Appearance F Table of Derivatives G Table of Integrals
A Worked-Out Solutions to Exercises: Chapters 1 to 9
Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9
Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 B Worked-Out Solutions to Exercises: Chapters 11 to 19
Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19
Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 C Worked-Out Solutions to Exercises: Chapters 21 to 29
Chapter 21 Chapter 22 Chapter 23 Chapter 24 Chapter 25 Chapter 26 Chapter 27 Chapter 28 Chapter 29
Chapter 21 Chapter 22 Chapter 23 Chapter 24 Chapter 25 Chapter 26 Chapter 27 Chapter 28 Chapter 29 D Answers to Final Exam Questions E Special Characters in Order of Appearance F Table of Derivatives G Table of Integrals Suggested Additional Reading Index (with page links)
A B C E F G H I JK L M N O P Q R S T U V W XYZ
A B C E F G H I JK L M N O P Q R S T U V W XYZ
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