Definite Integral - Limit of Riemann Sums

The Problem of Areas

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.

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

Course Material Related to This Topic:

• Read lecture notes, section 1 on pages 1–2

Partitions

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

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

Course Material Related to This Topic:

• Read lecture notes, section 2 on page 2

Riemann Sums

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

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

Course Material Related to This Topic:

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

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

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

Course Material Related to This Topic:

• Read chapter 20 of online textbook

The Riemann Integral

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

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

Course Material Related to This Topic:

• Read lecture notes, section 4 on pages 3–4

The Riemann Sum for the Exponential Function

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

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

Course Material Related to This Topic:

• Read lecture notes, section 1 on pages 1–2

The Riemann Sum for x^r

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

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

Course Material Related to This Topic:

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

Area and Notation

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

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

Course Material Related to This Topic:

• Read chapter 20 of online textbook

Evaluating a Riemann Integral

Using upper sums to evaluate a definite integral.

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

Course Material Related to This Topic:

• Complete practice problem 2 on page 2
• Check solution to practice problem 2 on page 3

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 problems 4.1–4.3 on page 3
• Check solution to exam problems 4.1–4.3 on page 3

Limit Definition of Integral

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

• 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 Jeriso

Course Material Related to This Topic:

• Complete exam problem 3B-1 on page 21 to Problem 3B-7 on page 22
• Check solution to exam problems on page 39