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:**

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:**

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:**

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:**

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:**

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:**

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:**

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:**

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

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

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

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