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CLP-2 Integral Calculus
Joel Feldman, Andrew Rechnitzer, Elyse Yeager
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Front Matter
Colophon
Preface
Dedication
Acknowledgements
Using the exercises in this book
Feedback about the text
1
Integration
1.1
Definition of the Integral
1.1.1
A Motivating Example
1.1.2
Optional — A more rigorous area computation
1.1.3
Summation notation
1.1.3.1
Proof of Theorem 1.1.6 (Optional)
1.1.4
The Definition of the Definite Integral
1.1.5
Using Known Areas to Evaluate Integrals
1.1.6
Another Interpretation for Definite Integrals
1.1.7
Optional — careful definition of the integral
1.1.8
Exercises
1.2
Basic properties of the definite integral
1.2.1
More properties of integration: even and odd functions
1.2.2
Optional — More properties of integration: inequalities for integrals
1.2.3
Exercises
1.3
The Fundamental Theorem of Calculus
1.3.1
The Fundamental Theorem of Calculus
1.3.2
Exercises
1.4
Substitution
1.4.1
Substitution
1.4.2
Exercises
1.5
Area between curves
1.5.1
Area between curves
1.5.2
Exercises
1.6
Volumes
1.6.1
Optional — Cylindrical shells
1.6.2
Exercises
1.7
Integration by parts
1.7.1
Integration by parts
1.7.2
Exercises
1.8
Trigonometric Integrals
1.8.1
Integrating
∫
sin
m
x
cos
n
x
d
x
1.8.1.1
One of
n
and
m
is odd
1.8.1.2
Both
n
and
m
are even
1.8.2
Integrating
∫
tan
m
x
sec
n
x
d
x
1.8.2.1
m
is odd — odd power of tangent
1.8.2.2
m
is odd and
n
≥
1
— odd power of tangent and at least one secant
1.8.2.3
n
≥
2
is even — a positive even power of secant
1.8.2.4
m
is even and
n
=
0
— even powers of tangent
1.8.3
Optional — Integrating
,
sec
x
,
,
csc
x
,
sec
3
x
and
csc
3
x
1.8.4
Exercises
1.9
Trigonometric Substitution
1.9.1
Trigonometric Substitution
1.9.2
Exercises
1.10
Partial Fractions
1.10.1
Partial fraction decomposition examples
1.10.2
The form of partial fraction decompositions
1.10.2.1
Simple linear factor case
1.10.2.2
General linear factor case
1.10.2.3
Simple linear and quadratic factor case
1.10.2.4
Optional — General linear and quadratic factor case
1.10.3
Optional — Justification of the partial fraction decompositions
1.10.3.1
Simple linear factor case
1.10.3.2
The general case with linear factors
1.10.3.3
Really Optional — The Fully General Case
1.10.4
Exercises
1.11
Numerical Integration
1.11.1
The midpoint rule
1.11.2
The trapezoidal rule
1.11.3
Simpson’s Rule
1.11.4
Three Simple Numerical Integrators — Error Behaviour
1.11.5
Optional — An error bound for the midpoint rule
1.11.6
Exercises
1.12
Improper Integrals
1.12.1
Definitions
1.12.2
Examples
1.12.3
Convergence Tests for Improper Integrals
1.12.4
Exercises
1.13
More Integration Examples
1.13
Exercises
2
Applications of Integration
2.1
Work
2.1.1
Work
2.1.2
Exercises
2.2
Averages
2.2.1
Optional — Return to the mean value theorem
2.2.2
Exercises
2.3
Centre of Mass and Torque
2.3.1
Centre of Mass
2.3.2
Optional — Torque
2.3.3
Exercises
2.4
Separable Differential Equations
2.4.1
Separate and integrate
2.4.2
Optional — Carbon Dating
2.4.3
Optional — Newton’s Law of Cooling
2.4.4
Optional — Population Growth
2.4.5
Optional — Mixing Problems
2.4.6
Optional — Interest on Investments
2.4.7
Exercises
3
Sequences and series
3.1
Sequences
3.1.1
Sequences
3.1.2
Exercises
3.2
Series
3.2.1
Series
3.2.2
Exercises
3.3
Convergence Tests
3.3.1
The Divergence Test
3.3.2
The Integral Test
3.3.3
The Comparison Test
3.3.4
The Alternating Series Test
3.3.5
The Ratio Test
3.3.6
Convergence Test List
3.3.7
Optional — The Leaning Tower of Books
3.3.8
Optional — The Root Test
3.3.9
Optional — Harmonic and Basel Series
3.3.9.1
The Harmonic Series
3.3.9.2
The Basel Problem
3.3.10
Optional — Some Proofs
3.3.11
Exercises
3.4
Absolute and Conditional Convergence
3.4.1
Definitions
3.4.2
Optional — The delicacy of conditionally convergent series
3.4.3
Exercises
3.5
Power Series
3.5.1
Radius and Interval of Convergence
3.5.2
Working With Power Series
3.5.3
Exercises
3.6
Taylor Series
3.6.1
Extending Taylor Polynomials
3.6.1
Optional — More about the Taylor Remainder
3.6.2
Computing with Taylor Series
3.6.3
Optional — Linking
e
x
with trigonometric functions
3.6.4
Evaluating Limits using Taylor Expansions
3.6.5
Optional — The Big O Notation
3.6.6
Optional — Evaluating Limits Using Taylor Expansions — More Examples
3.6.8
Exercises
3.7
Optional — Rational and irrational numbers
3.7.1
Decimal expansions of rational numbers
3.7.2
Eventually periodic implies rational
3.7.3
Rational implies eventually periodic
3.7.4
Irrationality of
e
3.7.4.1
Proof 1
3.7.4.2
Proof 2
3.7.5
Irrationality of
π
3.7.5.1
Bounding the integral
3.7.5.2
Integration by parts
3.7.5.3
The derivatives are integers
3.7.5.4
Putting it together
Appendices
A
High School Material
A.1
Similar Triangles
A.2
Pythagoras
A.3
Trigonometry — Definitions
A.4
Radians, Arcs and Sectors
A.5
Trigonometry — Graphs
A.6
Trigonometry — Special Triangles
A.7
Trigonometry — Simple Identities
A.8
Trigonometry — Add and Subtract Angles
A.9
Inverse Trigonometric Functions
A.10
Areas
A.11
Volumes
A.12
Powers
A.13
Logarithms
A.14
Highschool Material You Should be Able to Derive
A.15
Cartesian Coordinates
A.16
Roots of Polynomials
B
Complex Numbers and Exponentials
B.1
Definition and Basic Operations
B.2
The Complex Exponential
B.2.1
Definition and Basic Properties
B.2.2
Relationship with
sin
and
cos
B.2.3
Polar Coordinates
B.2.4
Exploiting Complex Exponentials in Calculus Computations
B.2.5
Exploiting Complex Exponentials in Differential Equation Computations
C
More About Numerical Integration
C.1
Richardson Extrapolation
C.2
Romberg Integration
C.3
Adaptive Quadrature
D
Numerical Solution of ODE’s
D.1
Simple ODE Solvers — Derivation
D.1.1
Euler’s Method
D.1.2
The Improved Euler’s Method
D.1.3
The Runge-Kutta Method
D.2
Simple ODE Solvers — Error Behaviour
D.2.1
Local Truncation Error for Euler’s Method
D.2.2
Global Truncation Error for Euler’s Method
D.3
Variable Step Size Methods
D.3.1
Euler and Euler-2step (preliminary version)
D.3.2
Euler and Euler-2step (final version)
D.3.3
Fehlberg’s Method
D.3.4
The Kutta-Merson Process
D.3.5
The Local Truncation Error for Euler-2step
E
Hints for Exercises
F
Answers to Exercises
G
Solutions to Exercises
🔗
Section
A.7
Trigonometry — Simple Identities
🔗
Periodicity
sin
(
θ
+
2
π
)
=
sin
(
θ
)
cos
(
θ
+
2
π
)
=
cos
(
θ
)
Reflection
sin
(
−
θ
)
=
−
sin
(
θ
)
cos
(
−
θ
)
=
cos
(
θ
)
Reflection around
π
/
4
sin
(
π
2
−
θ
)
=
cos
θ
cos
(
π
2
−
θ
)
=
sin
θ
Reflection around
π
/
2
sin
(
π
−
θ
)
=
sin
θ
cos
(
π
−
θ
)
=
−
cos
θ
Rotation by
π
sin
(
θ
+
π
)
=
−
sin
θ
cos
(
θ
+
π
)
=
−
cos
θ
Pythagoras
sin
2
θ
+
cos
2
θ
=
1
