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Lecture Notes
PDF
Section 1, Page 1
Introduction to approximation techniques other than vertical strips (Trapezoid Rule and Simpson's Rule).
Prof. Jason Starr
Riemann Integral (section 4 of lecture 14) and Fundamental Theorem of Calculus (section 3 of lecture 15)
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PDF
Section 2, Page 2
Derivation of the Trapezoid Rule for approximating Riemann integrals.
Prof. Jason Starr
Riemann Sums and Integrals (sections 3 and 4 of lecture 14)
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PDF
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.
Prof. Jason Starr
Riemann Sums and Integrals (sections 3 and 4 of lecture 14)
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Online Textbook Chapters
Document
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.
Prof. Daniel J. Kleitman
Riemann Sums (OT20.2)
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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.
Prof. Daniel J. Kleitman
Riemann Sums (OT20.2)
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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.
Prof. Daniel J. Kleitman
Trapezoid Rule (OT25.1)
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Document
Using extrapolation to greatly improve the accuracy of approximations using the Trapezoid Rule or Simpson's Rule.
Prof. Daniel J. Kleitman
Trapezoid Rule (OT25.1) and Simpsons Rule (OT25.2)
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Exam Questions
PDF
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.
Prof. Jason Starr
None
Solution (PDF) Pages 3 to 4
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PDF
Problem 4.1 (page 3) to problem 4.3 (page 3)
Three problems which involve evaluating Riemann sums and integrals.
Prof. Jason Starr
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Problem 9 (page 1)
Finding the approximate value of an integral using each rule with two subintervals.
Prof. David Jerison
None
Solution (PDF)# Page 1
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Problem 5 (page 1)
Estimating the number of hits a player got in a month using the two rules.
Prof. David Jerison
None
Solution (PDF)# Page 1
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Problem 2 (page 1)
Using Riemann Sums, the Trapezoid Rule, and Simpson's Rule to approximate a definite integral.
Prof. David Jerison
None
Solution (PDF)# Page 1
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Problem 9 (page 2)
Estimating a definite integral of the sine-squared function using three intervals of the Trapezoidal Rule.
Prof. David Jerison
None
Solution (PDF)# Page 1
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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.
Prof. David Jerison
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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.
Prof. David Jerison
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Java Applet
Java Applet
Requires Java Virtual Machine
Applet which uses the Left Hand Rule, Right Hand Rule, Trapezoid Rule, and Simpson's Rule to approximate the area under a specified curve.
Prof. Daniel J. Kleitman
Special Riemann Sums (OT20.5), Trapezoid Rule (OT25.1), and Simpsons Rule (OT25.2)
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