Improper Integrals

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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

Comparison Tests for Convergence

Suppose that

$f(x)$ and

$g(x)$ are functions on

$[a,\infty)$ , and that

$0 \le f(x) \le g(x) \quad\text{ for all }x \in [a,\infty).$ Then we say

$f$ is bounded between

$0$ and

$g$ . Observe the following, based on the inequality above:

If $\int_a^\infty g(x)\, dx$ converges, then $\int_a^\infty f(x)\, dx$ must also converge.

If $\int_a^\infty f(x)\, dx$ diverges, then $\int_a^\infty g(x)\, dx$ must also diverge.

If $\int_a^\infty f(x) \,dx$ converges, we know nothing about $\int_a^\infty g(x) \,dx$ - it might converge or diverge.

If $\int_a^\infty g(x) \,dx$ diverges, we know nothing about $\int_a^\infty f(x) \,dx$ - it might converge or diverge.

Example:

$\displaystyle\int_1^\infty \frac{dx}{x^4}$ converges, and, since

$\displaystyle\frac{1}{x^4+1} < \frac{1}{x^4}$ , we can conclude that also

$\displaystyle\int_1^\infty \frac{dx}{1+x^4}$ converges. Notice that the test doesn't tell us the value of the integral, just that it converges.

The idea of comparison tests will be used often when we get to sequences and series. It's a good idea to pay close attention to the following video, and get a feel for comparison tests now.

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