# Improper Integrals

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## Lecture Notes

#### A Problem With Riemann Integrals

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Section 1, Page 1 to page 2

Explanation that Riemann integrals are not defined when the interval is unbounded but can often be found using limits. Mention of the alternative Lebesgue integral.

Instructor: Prof. Jason Starr
Prior Knowledge: Riemann Integrals (section 4 of lecture 14).

#### Improper Integrals of the First Kind

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Section 2, Page 2 to page 3

Using limits to evaluate improper integrals with unbounded limits of integration. Includes examples of integrating 1/(xp) from 1 to infinity and integrating cos(x) from 0 to infinity.

Instructor: Prof. Jason Starr
Prior Knowledge: Limits (section 2 of lecture 2) and Riemann Integrals (section 4 of lecture 14).

#### Improper Integrals of the Second Kind

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Section 3, Page 3 to page 4

Using limits to evaluate improper integrals involving functions that are unbounded over the specified limits of integration. Includes example of integrating 1/(xp) between 0 and 1.

Instructor: Prof. Jason Starr
Prior Knowledge: Limits (section 2 of lecture 2) and Riemann Integrals (section 4 of lecture 14).

#### The Comparison Test

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Section 4, Page 4 to page 5

Definition of monotone bounded limits, the squeezing lemma for limits and improper integrals, and the comparison test for convergence of improper integrals.

Instructor: Prof. Jason Starr
Prior Knowledge: Improper Integrals (sections 2 and 3 of this lecture)

#### Improper Integrals

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Page 1 to page 3

The Comparison Test for determining convergence or divergence of improper integrals, with discussion and examples.

Instructor: Prof. David Jerison
Prior Knowledge: None

## Exam Questions

#### Improper Integral

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Problem 4 (page 1)

Determining whether an improper integral converges or diverges.

Instructor: Prof. Jason Starr
Prior Knowledge: None

#### Improper Integrals

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Problem 8.1 (page 6) to problem 8.2 (page 6)

Two questions which involve determining whether an improper integral will converge or diverge.

Instructor: Prof. Jason Starr
Prior Knowledge: None

#### Improper Integrals

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Problem 16 (page 2)

An integral with an infinite upper limit of integration to be evaluated.

Instructor: Prof. David Jerison
Prior Knowledge: None