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$x=a$

Representing Functions as Power Series

Functions as Power Series
Derivatives and Integrals of Power Series
Applications and Examples

Taylor and Maclaurin Series

The Formula for Taylor Series
Taylor Series for Common Functions
Adding, Multiplying, and Dividing Power Series
Miscellaneous Useful Facts

Applications of Taylor Polynomials

Taylor Polynomials
When Functions Are Equal to Their Taylor Series
When a Function Does Not Equal Its Taylor Series
Other Uses of Taylor Polynomials

Partial Derivatives

Visualizing Functions in 3 Dimensions
Definitions and Examples
An Example from DNA
Geometry of Partial Derivatives
Higher Order Derivatives
Differentials and Taylor Expansions

Multiple Integrals

Background
What is a Double Integral?
Volumes as Double Integrals

Iterated Integrals over Rectangles

How To Compute Iterated Integrals
Examples of Iterated Integrals
Cavalieri's Principle
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
Order of Integration

Type I and Type II regions

If $R$ is a rectangle in the $x$ - $y$ plane and $f(x,y)$ is a function defined on $R$ then we saw that $\displaystyle\iint_R f(x,y)\, dA$ is what we get when we

Chop $R$ into a bunch of small boxes.
Compute $f\left(x^*,y^*\right)\,\Delta x\, \Delta y$ for each box contained in $R$ .
Add up the boxes, and
Take a limit as we chop $R$ into smaller and smaller pieces.

The same ideas apply when $R$ is an arbitrary region that is not a rectangle. As with rectangles, things get a lot simpler if we arrange the boxes intelligently. This video explains the concepts.

The video above explained why we get the following summary of integrals over an arbitrary region. Remember, $g$ and $h$ are functions on the $xy$ -plane, while $f(x,y)$ is a surface over (or under) the $xy$ -plane.

Type I regions are regions that are bounded by vertical lines $x=a$ and $x=b$ , and curves $y=g(x)$ and $y=h(x)$ , where we assume that $g(x) < h(x)$ and $a < b$ . Then we can integrate first over $y$ and then over $x$ : $\iint_R f(x,y)\, dA = \int_{x=a}^b\, \int_{y=g(x)}^{h(x)} \,f(x,y)\, dy \, dx$ instruct.math.lsa.umich.edu
Type II regions are bounded by horizontal lines $y=c$ and $y=d$ , and curves $x=g(y)$ and $x=h(y)$ , where we assume that $g(y)< h(y)$ and $c < d$ . Then we can integrate first over $x$ and then over $y$ : $\iint_R f(x,y)\, dA = \int_{y=c}^d \,\int_{x=g(y)}^{h(y)} \,f(x,y)\, dx\, dy$ instruct.math.lsa.umich.edu

The Fundamental Theorem of Calculus

The Indefinite Integral and the Net Change

Substitution

Area Between Curves

Volumes

Integration by Parts

Integrals of Trig Functions

Trig Substitutions

Partial Fractions

Strategies of Integration

Improper Integrals

Differential Equations

Models of Growth

Infinite Sequences

Infinite Series

Integral Test

Comparison Tests

Convergence of Series with Negative Terms

The Ratio and Root Tests

Strategies for testing Series

Power Series