Riemann & Trapezoidal Sums

Approximating Riemann Integrals

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

• 18.01 Single Variable Calculus, Fall 2005
  Prof. Jason Starr

Course Material Related to This Topic:

• Read lecture notes, section 1 on page 1

The Trapezoid Rule

Derivation of the Trapezoid Rule for approximating Riemann integrals.

• 18.01 Single Variable Calculus, Fall 2005
  Prof. Jason Starr

Course Material Related to This Topic:

• Read lecture notes, section 2 on page 2

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.

• 18.013A Calculus with Applications, Spring 2005
  Prof. Daniel J. Kleitman

Course Material Related to This Topic:

• Read chapter 25 of online textbook

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

• 18.01 Single Variable Calculus, Fall 2006
  Prof. David Jerison

Course Material Related to This Topic:

• Complete exam problem 9 on page 2
• Check solution toexam problem 9 on page 1

Simpson's Rule

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

• 18.01 Single Variable Calculus, Fall 2005
  Prof. Jason Starr

Course Material Related to This Topic:

• Read lecture notes, section 3 on pages 2–5

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.

• 18.013A Calculus with Applications, Spring 2005
  Prof. Daniel J. Kleitman

Course Material Related to This Topic:

• Read chapter 25 of online textbook

Special Riemann Sums

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.

• 18.013A Calculus with Applications, Spring 2005
  Prof. Daniel J. Kleitman

Course Material Related to This Topic:

• Read chapter 20 of online textbook

Extrapolations and Better Approximations

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

• 18.013A Calculus with Applications, Spring 2005
  Prof. Daniel J. Kleitman

Course Material Related to This Topic:

• Read chapter 25 of online textbook

Riemann Sum

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

• 18.01 Single Variable Calculus, Fall 2005
  Prof. Jason Starr

Course Material Related to This Topic:

• Complete exam problem 2 on pages 3–4
• Check solution to exam problem 2 on pages 3–4

Riemann Sums and Integrals

Three problems which involve evaluating Riemann sums and integrals.

• 18.01 Single Variable Calculus, Fall 2005
  Prof. Jason Starr

Course Material Related to This Topic:

• Complete exam problem 4.1 to 4.3 on page 3
• Check solution to exam problems 4.1 to 4.3 on page 3

Trapezoid Rule and Simpsons Rule

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

• 18.01 Single Variable Calculus, Fall 2006
  Prof. David Jerison

Course Material Related to This Topic:

• Complete exam problem 9 on page 1
• Check solution to exam problem 9 on page 1

Trapezoid Rule and Simpson's Rule: Baseball Stats

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

• 18.01 Single Variable Calculus, Fall 2006
  Prof. David Jerison

Course Material Related to This Topic:

• Complete exam problem 5 on page 1
• Check solution to exam problem 5 on page 1

Numeric Approximations of Integrals

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

• 18.01 Single Variable Calculus, Fall 2006
  Prof. David Jerison

Course Material Related to This Topic:

• Complete exam problem 2 on page 1
• Check solution to exam problem 2 on page 1

Definite Integrals

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

• 18.01 Single Variable Calculus, Fall 2006
  Prof. David Jerison

Course Material Related to This Topic:

• Complete exam problems 3B-1 to 3B-7 on pages 21–2
• Check solution to exam problems 3B-1 to 3B-7 on page 39

Numerical Integration

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

• 18.01 Single Variable Calculus, Fall 2006
  Prof. David Jerison

Course Material Related to This Topic:

• Complete exam problems 3G-1 to 3G-5 on pages 26–7
• Check solution to exam problems on 3G-1 to 3G-5 pages 47–9

Applet which uses the Left Hand Rule, Right Hand Rule, Trapezoid Rule, and Simpson's Rule to approximate the area under a specified curve.

• 18.013A Calculus with Applications, Spring 2005
  Prof. Daniel J. Kleitman

Course Material Related to This Topic:

• Interact with a Java Simulation