Arc Length and Surface Area

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Integration by Parts

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Integration by Parts with a definite integral
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Trig Substitutions

How Trig Substitution Works
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Completing the Square

Partial Fractions

Introduction to Partial Fractions
Linear Factors
Irreducible Quadratic Factors
Improper Rational Functions and Long Division
Summary

Strategies of Integration

Substitution
Integration by Parts
Trig Integrals
Trig Substitutions
Partial Fractions

Improper Integrals

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Comparison Tests for Convergence

Modeling with Differential Equations

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Separable Equations
A Second Order Problem

Euler's Method and Direction Fields

Euler's Method (follow your nose)
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Euler's method revisited

Separable Equations

The Simplest Differential Equations
Separable differential equations
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Models of Growth

Exponential Growth and Decay
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Linear Equations

Linear ODEs: Working an Example
The Solution in General
Saving for Retirement

Parametrized Curves

Three kinds of functions, three kinds of curves
The Cycloid
Visualizing Parametrized Curves
Tracing Circles and Ellipses
Lissajous Figures

Calculus with Parametrized Curves

Video: Slope and Area
Video: Arclength and Surface Area
Summary and Simplifications
Higher Derivatives

Polar Coordinates

Definitions of Polar Coordinates
Graphing polar functions
Video: Computing Slopes of Tangent Lines

Areas and Lengths of Polar Curves

Area Inside a Polar Curve
Area Between Polar Curves
Arc Length of Polar Curves

Conic sections

Slicing a Cone
Ellipses
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Shifting the Center by Completing the Square

Conic Sections in Polar Coordinates

Foci and Directrices
Visualizing Eccentricity
Astronomy and Equations in Polar Coordinates

Infinite Sequences

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Infinite Series

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Telescoping Sums

Integral Test

Preview of Coming Attractions
The Integral Test
Estimates for the Value of the Series

Comparison Tests

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The Limit Comparison Test

Convergence of Series with Negative Terms

Introduction, Alternating Series,and the AS Test
Absolute Convergence
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The Ratio and Root Tests

The Ratio Test
The Root Test
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Strategy to Test Series and a Review of Tests
Examples, Part 1
Examples, Part 2

Power Series

Radius and Interval of Convergence
Finding the Interval of Convergence
Power Series Centered at

$x=a$

Representing Functions as Power Series

Functions as Power Series
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Taylor and Maclaurin Series

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Taylor Polynomials
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Functions of 2 and 3 variables

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Partial Derivatives

One variable at a time (yet again)
Definitions and Examples
An Example from DNA
Geometry of partial derivatives
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Differentiability and the Chain Rule

Differentiability
The First Case of the Chain Rule
Chain Rule, General Case
Video: Worked problems

Multiple Integrals

General Setup and Review of 1D Integrals
What is a Double Integral?
Volumes as Double Integrals

Iterated Integrals over Rectangles

How To Compute Iterated Integrals
Examples of Iterated Integrals
Fubini's Theorem
Summary and an Important Example

Double Integrals over General Regions

Type I and Type II regions
Examples 1-4
Examples 5-7
Swapping the Order of Integration
Area and Volume Revisited

Double integrals in polar coordinates

dA = r dr (d theta)
Examples

Multiple integrals in physics

Double integrals in physics
Triple integrals in physics

Integrals in Probability and Statistics

Single integrals in probability
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Change of Variables

Review: Change of variables in 1 dimension
Mappings in 2 dimensions
Jacobians
Examples
Bonus: Cylindrical and spherical coordinates

Next we want to figure out the length of a parametrized curve. As with all integrals, we break it into pieces, estimate each piece, add the pieces together, and take a limit.

In the following video we derive the formulas for these quantities, compute the arclength of a cycloid, and compute the surface area of the associated surface of revolution.

Summary: If we have a parametrized curve running from time

$t_1$ to time

$t_2$ , then

The arc length of the curve is $L = \int_{t_1}^{t_2} \sqrt{\left ( \frac{dx}{dt}\right )^2 + \left ( \frac{dy}{dt}\right )^2} \;dt,$
If we rotate the curve around the $x$ axis we get a surface, called a surface of revolution. The area of this surface is $\int_{t_1}^{t_2} 2 \pi y \; \sqrt{\left ( \frac{dx}{dt}\right )^2 + \left ( \frac{dy}{dt}\right )^2} \;dt.$