CLP-2 Integral Calculus

Integration
Definition of the Integral
Optional — A More Rigorous Area Computation
Summation Notation
The Definition of the Definite Integral
Using Known Areas to Evaluate Integrals
Another Interpretation for Definite Integrals
Optional — Careful Definition of the Integral
Basic Properties of the Definite Integral
More Properties of Integration: Even and Odd Functions
Optional — More Properties of Integration: Inequalities for Integrals
The Fundamental Theorem of Calculus
Substitution
Area Between Curves
Volumes
Integration by Parts
Trigonometric Integrals
Integrating sinm xcosn xdx
Integrating tanm xsecn xdx
Optional — integrating secx, cscx, sec3 x and csc3 x
Trigonometric Substitution
Partial Fractions
Partial Fraction Decomposition Examples
The Form of Partial Fraction Decompositions
Optional — Justification of the Partial Fraction Decompositions
Numerical Integration
The Midpoint Rule
The Trapezoidal Rule
Simpson's Rule
Three Simple Numerical Integrators – Error Behaviour
Optional — An Error Bound for the Midpoint Rule
Improper Integrals
Definitions
Examples
Convergence Tests for Improper Integrals
Applications of Integration
Work
Averages
Centre of Mass and Torque
Centre of Mass
Optional — Torque
Separable Differential Equations
Separate and Integrate
Optional — Carbon Dating
Optional — Newton's Law of Cooling
Optional — Population Growth
Optional — Mixing Problems
Optional — Interest on Investments
Sequence and Series
Sequences
Series
Convergence Tests
The Divergence Test
The Integral Test
The Comparison Test
The Alternating Series Test
The Ratio Test
Convergence Test List
Optional — The Leaning Tower of Books
Optional — The Root Test
Optional — Harmonic and Basel Series
Optional — Some Proofs
Absolute and Conditional Convergence
Definitions
Optional — The Delicacy of Conditionally Convergent Series
Power Series
Radius and Interval of Convergence
Working With Power Series
Taylor Series
Extending Taylor Polynomials
Computing with Taylor Series
Optional — Linking ex with Trigonometric Functions
Evaluating Limits using Taylor Expansions
Optional — The Big O Notation
Optional — Evaluating Limits Using Taylor Expansions — More Examples
Optional — Rational and Irrational Numbers
High school material
Similar Triangles
Pythagoras
Trigonometry — Definitions
Radians, Arcs and Sectors
Trigonometry — Graphs
Trigonometry — Special Triangles
Trigonometry — Simple Identities
Trigonometry — Add and Subtract Angles
Inverse Trigonometric Functions
Areas
Volumes
Powers
Logarithms
Highschool Material You Should be Able to Derive
Cartesian Coordinates
Roots of Polynomials
Complex Numbers and Exponentials
Definition and Basic Operations
The Complex Exponential
Definition and Basic Properties.
Relationship with sin and cos.
Polar Coordinates.
Exploiting Complex Exponentials in Calculus Computations
Exploiting Complex Exponentials in Differential Equation Computations
More About Numerical Integration
Richardson Extrapolation
Romberg Integration
Adaptive Quadrature
Numerical Solution of ODE's
Simple ODE Solvers — Derivation
Euler's Method
The Improved Euler's Method
The Runge-Kutta Method
Simple ODE Solvers — Error Behaviour
Local Truncation Error for Euler's Method
Global Truncation Error for Euler's Method
Variable Step Size Methods
Euler and Euler-2step (preliminary version)
Euler and Euler-2step (final version)
Fehlberg's Method
The Kutta-Merson Process
The Local Truncation Error for Euler-2step