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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Integrals of Trig Functions

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Partial Fractions

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

Strategies of Integration

Substitution
Integration by Parts
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Trig Substitutions
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Improper Integrals

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Comparison Tests for Convergence

Modeling with Differential Equations

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A Second Order Problem

Euler's Method and Direction Fields

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

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Exponential Growth and Decay
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Linear Equations

Linear ODEs: Working an Example
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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
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Arc Length of Polar Curves

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Conic Sections in Polar Coordinates

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Astronomy and Equations in Polar Coordinates

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Infinite Series

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Integral Test

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The Integral Test
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Convergence of Series with Negative Terms

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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
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Power Series

Radius and Interval of Convergence
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Representing Functions as Power Series

Functions as Power 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)
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An Example from DNA
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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?
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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 $$\int_{t_1}^{t_2} 2 \pi y \; \sqrt{\left ( \frac{dx}{dt}\right )^2 + \left ( \frac{dy}{dt}\right )^2} \;dt.$$