Overview

Dive into a comprehensive 7-hour Calculus II course covering integration methods, series, parametric and polar coordinates, and vectors. Master integration by parts, trigonometric integrals, and trigonometric substitutions. Explore improper integrals, arc length, and surface area of revolution. Delve into sequences and series, including power series, Taylor series, and convergence tests. Learn about vectors and planes. Tackle challenging examples and gain insights into choosing appropriate integration methods. Understand the hierarchy of big functions, explore the bizarre world of infinite rearrangements, and discover applications of Taylor series. Investigate parametric curves, polar coordinates, and complex numbers. Enhance your mathematical skills with Dr. Trefor Bazett's expert guidance through this extensive Calculus II curriculum.

Syllabus

Welcome to Calculus II.
Intro to Integration By Parts --- Examples: ∫xsinxdx & ∫arctan(x)dx.
Two Tricky Integration By Parts Examples.
Trigonometric Integrals --- ∫sin^n(x)cos^m(x)dx via Pythagorean or Half-Angle Identities.
Intro to Trigonometric Substitution --- Ex: Deriving Area of Circle Formula.
Trig Subs | How To Choose The Substitution & Deal With Indefinite Integrals.
Deconstructing a messy integral | Trig subs & u-subs combined.
Integration by Partial Fractions | Big Idea + First Example.
Partial Fractions | Repeating and irreducible Quadratic Terms.
Choosing what integration methods to use - Part I.
Choosing what integration methods to use - Challenging Example!.
MATH PROF vs TRICKY INTEGRALS.
Improper Integrals: How to Integrate with Infinities, 2 ways!.
Comparison Test for Improper Integrals.
Arclength Formula | Derivation & Ex: Circumference of a Circle.
Area of Surfaces of Revolution | Derivation & Example.
Infinite Surface Area but Finite Volume!?!? *Gabriel's Horn*.
Intro to Sequences.
Limits of Sequences, Limit laws & Function Representations.
The Hierarchy of Big Functions || n^n greater than n! greater than e^n greater than n^100.
Intro to Series: What is 1/2+1/4+1/8+1/16+...?.
Geometric Series | Convergence, Derivation, and Example.
Harmonic Series | It diverges, but insanely slowly!.
Integral Test | Derivation & 1st Example.
Estimating the Remainder of a Series Approximation via the Integral Test.
Comparison Test for Series.
Limit Comparison Test for Series.
Alternating Series Test | Intuition, Statement & Example.
Alternating Series | Estimating the Remainder.
The bizarre world of INFINITE rearrangements // Riemann Series Theorem.
Absolute Convergence vs Conditional Convergence vs Convergence.
Ratio & Root Tests | Geometric Series Generalized.
Choosing Which Convergence Test to Apply to 8 Series.
Solving Inequalities with Absolute Values: Ex |x-2| less than 3.
Power Series & Intervals of Convergence.
Intro to Taylor Series: Approximations on Steroids.
3 Applications of Taylor Series: Integrals, Limits, & Series.
Why Taylor Series actually work: The Taylor Inequality.
Plotting Parametric Curves.
Tangents to Parametric Curves | Multiple tangents at the same point!!.
Area under a Parametric Curve | Formula, Derivation, & Example.
Arclength of Parametric Curves.
Intro to Polar Coordinates.
Sketching Polar Curves.
Areas in Polar Coordinates.
Example: Area Inside Polar Curves.
Why Exponential Growth?? Intro to Separable Differential Equations.
Defining the Natural Logarithm as an Integral?!?!?.
Intro to COMPLEX NUMBERS // Motivation, Algebraic Definition & Fundamental Theorem of Algebra Ep. 1.
The geometric view of COMPLEX NUMBERS.
The Polar Form of COMPLEX NUMBERS // Finding the nth roots of -1.