# Sigma Notation and Riemann Sums

## Area Under A Curve – Riemann Sums

Suppose we want to calculate the area between the graph of a positive function and the interval on the x–axis (Fig. 7). The Riemann Sum method is to build several rectangles with bases on the interval and sides that reach up to the graph
of f (Fig. 8). Then the areas of the rectangles can be calculated and added together to get a number called a Riemann Sum of f on [a, b]. The area of the region formed by the rectangles is an
** approximation **of the area we want.

**Example 5: **Approximate the area in Fig. 9a between the graph of and the interval on the x–axis by summing the areas of the rectangles in Fig. 9b.

Solution: The total area of rectangles is square units.

In order to effectively describe this process, some new vocabulary is helpful: a **partition** of an interval and the **mesh** of the partition.

A **partition** of a closed interval into subintervals is a set of points in increasing order, .

(A partition is a collection of points on the axis and it does not depend on the function in any way.)

The points of the partition divide the interval into subintervals (Fig. 10): with lengths , and . The points of the partition P are the locations of the vertical lines for the sides
of the rectangles, and the bases of the rectangles have lengths .

The **mesh **or** norm **of partition is the length of the longest of the subintervals , or, equivalently, the maximum of .

For example, the set is a partition of the interval (Fig. 11) and divides the interval into 4 subintervals with lengths . The mesh of this partition is 1.6, the maximum of the lengths of the subintervals. (If the mesh of a partition is "small," then the length of each one of the subintervals is the same or smaller.)

**Practice 6: ** is a partition of what interval? How many subintervals does it create? What is the mesh of the partition? What are the values of and ?

A function, a partition, and a point in each subinterval determine a Riemann sum.

Suppose is a positive function on the interval

is an x–value in the k^{th} subinterval .

Then the area of the k^{th} rectangle is . (Figure 12)

**Definition: **A summation of the form is called a** Riemann Sum **of for the partition .

This Riemann sum is the total of the areas of the rectangular regions and is an approximation of the area between the graph of f and the x–axis.

**Example 6:**

Find the Riemann sum for and the partition using the values and . (Fig. 13)

Solution: The 2 subintervals are and so and .

Then the Riemann sum for this partition is

**Practice 7:**

**Practice 8:** What is the smallest value a Riemann sum for and the partition can have? (You will need to select values for and .) What is the largest value a Riemann sum can have for this function
and partition?

Table 2 shows the results of a computer program that calculated Riemann sums for the function with different numbers of subintervals and different ways of selecting the points in each subinterval.

When the mesh of the partition is small (and the number of subintervals large), all of the ways of selecting the lead to approximately the same number for the Riemann sums. For this decreasing function, using the left endpoint of the subinterval always resulted in a sun that was larger than the area. Choosing the right end point gave a value smaller that the area. Why?

Table 2: Riemann sums for on the interval

Values of the Riemann sum for different choices of

As the mesh gets smaller, all of the Riemann Sums seem to be approaching the same value, approximately .

**Example 7:** Find the Riemann sum for the function on the interval using the partition with .

Solution: The 3 subintervals (Fig. 14) are , and so and . The Riemann sum for this partition is

**Practice 9:** Find the Riemann sum for the function and partition in the previous example, but use .