# Definite Integral - Limit of Riemann Sums

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## Lecture Notes

#### The Problem of Areas

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Section 1, Page 1 to page 2

Riemann integrals are introduced as a concept using the example of finding the area of a circle from the areas of N-sided polygons inscribed in the circle. Signed area (positive above the x-axis, negative below) is introduced.

Instructor: Prof. Jason Starr
Prior Knowledge: None

#### Partitions

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Section 2, Page 2

Interval partitions are defined, including the concepts of mesh size and fine vs. coarse partitions.

Instructor: Prof. Jason Starr
Prior Knowledge: None

#### Riemann Sums

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Section 3, Page 2 to page 3

Definition, including a discussion of partition choices when computing these sums.

Instructor: Prof. Jason Starr
Prior Knowledge: Partitions (section 2 of this lecture)

#### The Riemann Integral

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Section 4, Page 3 to page 4

Definite integrals are defined. Includes an example using the function f(x) = x.

Instructor: Prof. Jason Starr
Prior Knowledge: Partitions and Riemann Sums (sections 2 and 3 of this lecture)

#### The Riemann Sum for the Exponential Function

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Section 1, Page 1 to page 2

Using Riemann sums to find the Riemann integral for the function f(x) = ex.

Instructor: Prof. Jason Starr
Prior Knowledge: Riemann Sums and Integrals (lecture 14)

#### The Riemann Sum for xr

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Section 2, Page 3 to page 5

Using Riemann sums to find the Riemann integral for the function f(x) = xr.

Instructor: Prof. Jason Starr
Prior Knowledge: Riemann Sums and Integrals (lecture 14)

## Online Textbooks

#### Area and Notation

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Definition of the definite integral as the area under a curve, including definition of the integrand.

Instructor: Prof. Daniel J. Kleitman
Prior Knowledge: None

#### Riemann Sums

Document

Definition, including the use of Riemann sums in finding the area under a curve.

Instructor: Prof. Daniel J. Kleitman
Prior Knowledge: Area Under a Curve (OT20.1)

## Practice Problems

#### Evaluating a Riemann Integral

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Problem 2 (page 2)

Using upper sums to evaluate a definite integral.

Instructor: Prof. Jason Starr
Prior Knowledge: None

## Exam Questions

#### Riemann Sums and Integrals

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Problem 4.1 (page 3) to Problem 4.3 (page 3)

Three problems which involve evaluating Riemann sums and integrals.

Instructor: Prof. Jason Starr
Prior Knowledge: None

#### Limit Definition of Integral

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Problem 2 (page 1)

Evaluating an integral using the definition of an integral as the limit of sums.

Instructor: Prof. David Jerison
Prior Knowledge: None