In this section, we will begin to look at Σ notation and how it can be used to represent Riemann sums (rectangle approximations) of definite integrals.

Section 5.7 Working With Sums

Definition

Summation or Sigma notation is defined by:

Example 1

Find each of the following sums:

(a)

(b)

(c)

Some Special Sums

The following are sums with which we will need to work:

Example 2

(a) Use sigma notation to express R10 for and then evaluate it.

(b)Use sigma notation to express L20 for and then evaluate it.

Definition

Recall that the definite integral can be defined as a limit of sums:

where the ck are determined by whether we are using left, right, or midpoint rectangles.

Example 3

(a) Give the summation notation of Rn for and simplify the result.

(b)Use the limit definition of the definite integral to evaluate .

Example 4

(a) Give the summation notation of Rn for and simplify the result.

(b)Use the limit definition of the definite integral to evaluate .

Example 5

Evaluate the indicated limit by rewriting it as a definite integral and using the F.T.C.

Example 6

Evaluate the indicated limit by rewriting it as a definite integral and using the F.T.C.