# Sigma Notation and Riemann Sums

## Sigma Notation and Riemann Sums

One strategy for calculating the area of a region is to cut the region into simple shapes, calculate the area of each simple shape, and then add these smaller areas together to get the area of the whole region. We will use that approach, but it is useful to have a notation for adding a lot of values together: the sigma notation.

The variable (typically , , or ) used in the summation is called the **counter** or **index variable**. The function to the right of the sigma is called the **summand,** and the numbers below and above the sigma
are called the** lower and upper limits of the summation**. (Fig. 1)

**Practice 1: **Write the summation denoted by each of the following:

In practice, the sigma notation is frequently used with the standard function notation:

1 | 2 | 4 | 3 |

2 | 3 | 1 | 3 |

3 | 1 | -2 | 3 |

4 | 0 | 3 | 3 |

5 | 3 | 5 | 3 |

Table 1

**Example 1**: Use the values in Table 1 to evaluate and .

**Practice 2: **Use the values of , and in Table 1 to evaluate the following:

**Practice 3:** For , evaluate and .

The summand does not have to contain the index variable explicitly: a sum from to of the constant function can be written as

Since the sigma notation is simply a notation for addition, it has all of the familiar properties of addition

##### Summation Properties

Problems 16 and 17 illustrate that similar patterns for sums of products and quotients are **not valid**.

Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-5.2-Sigma-Notation-and-Riemann-Sums.pdf

This work is licensed under a Creative Commons Attribution 3.0 License.