# Riemann & Trapezoidal Sums

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## Lecture Notes

#### Approximating Riemann Integrals

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Section 1, Page 1

Introduction to approximation techniques other than vertical strips (Trapezoid Rule and Simpson's Rule).

Instructor: Prof. Jason Starr
Prior Knowledge: Riemann Integral (section 4 of lecture 14) and Fundamental Theorem of Calculus (section 3 of lecture 15)

#### The Trapezoid Rule

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Section 2, Page 2

Derivation of the Trapezoid Rule for approximating Riemann integrals.

Instructor: Prof. Jason Starr
Prior Knowledge: Riemann Sums and Integrals (sections 3 and 4 of lecture 14)

#### Simpson's Rule

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Section 3, Page 2 to page 5.

Derivation of Simpson's Rule for approximating Riemann integrals. Worked example using Trapezoid Rule and Simpson's Rule.

Instructor: Prof. Jason Starr
Prior Knowledge: Riemann Sums and Integrals (sections 3 and 4 of lecture 14)

## Online Textbook Chapters

#### Special Riemann Sums

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Finding Riemann sums with fixed widths using the leftmost, rightmost, maximum, and minimum argument in each strip, including comparisons of each and an applet for finding the leftmost and rightmost Riemann sums for a function.

Instructor: Prof. Daniel J. Kleitman
Prior Knowledge: Riemann Sums (OT20.2)

#### Trapezoid Rule

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Definition of this rule for approximating the area under a curve, including a measure of the error for this method compared to the actual value of the area.

Instructor: Prof. Daniel J. Kleitman
Prior Knowledge: Riemann Sums (OT20.2)

#### Simpson's Rule

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Definition and formula for this rule which uses quadratics to approximate the area under a curve, including a comparison of this and the Trapezoid Rule. Also includes an applet for finding the area under a curve using the rectangular left, rectangular right, trapezoid, and Simpson's Rule.

Instructor: Prof. Daniel J. Kleitman
Prior Knowledge: Trapezoid Rule (OT25.1)

#### Extrapolations and Better Approximations

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Using extrapolation to greatly improve the accuracy of approximations using the Trapezoid Rule or Simpson's Rule.

Instructor: Prof. Daniel J. Kleitman
Prior Knowledge: Trapezoid Rule (OT25.1) and Simpsons Rule (OT25.2)

## Exam Questions

#### Riemann Sum

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Problem 2 (Page 3 to page 4)

Computing the right endpoint Riemann sum of an integral and then using that answer to evaluate a limit.

Instructor: Prof. Jason Starr
Prior Knowledge: None

#### 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

#### Trapezoid Rule and Simpsons Rule

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Problem 9 (page 1)

Finding the approximate value of an integral using each rule with two subintervals.

Instructor: Prof. David Jerison
Prior Knowledge: None

#### Trapezoid Rule and Simpson's Rule: Baseball Stats

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Problem 5 (page 1)

Estimating the number of hits a player got in a month using the two rules.

Instructor: Prof. David Jerison
Prior Knowledge: None

#### Numeric Approximations of Integrals

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Problem 2 (page 1)

Using Riemann Sums, the Trapezoid Rule, and Simpson's Rule to approximate a definite integral.

Instructor: Prof. David Jerison
Prior Knowledge: None

#### Trapezoidal Rule

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Problem 9 (page 2)

Estimating a definite integral of the sine-squared function using three intervals of the Trapezoidal Rule.

Instructor: Prof. David Jerison
Prior Knowledge: None

#### Definite Integrals

PDF - 2.2 MB
Problem 3B-1 (page 21) to problem 3B-7 (page 22)

Seven questions which involve using sigma notation for sums, computing Riemann sums for definite integrals, and evaluating limits by relating them to Riemann sums.

Instructor: Prof. David Jerison
Prior Knowledge: None

#### Numerical Integration

PDF - 2.2 MB
Problem 3G-1 (page 26) to problem 3G-5 (page 27)

Five questions which involve approximating integrals using Riemann sums, the Trapezoidal Rule, and Simpson's Rule.

Instructor: Prof. David Jerison
Prior Knowledge: None