In this section, we will begin to look at notation and how it can be used to represent Riemann sums...
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![Page 1: In this section, we will begin to look at notation and how it can be used to represent Riemann sums (rectangle approximations) of definite integrals.](https://reader038.fdocuments.us/reader038/viewer/2022100505/5a4d1b8e7f8b9ab0599bfe80/html5/thumbnails/1.jpg)
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
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Definition
Summation or Sigma notation is defined by:
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Example 1
Find each of the following sums:
(a)
(b)
(c)
![Page 4: In this section, we will begin to look at notation and how it can be used to represent Riemann sums (rectangle approximations) of definite integrals.](https://reader038.fdocuments.us/reader038/viewer/2022100505/5a4d1b8e7f8b9ab0599bfe80/html5/thumbnails/4.jpg)
Some Special Sums
The following are sums with which we will need to work:
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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.
![Page 6: In this section, we will begin to look at notation and how it can be used to represent Riemann sums (rectangle approximations) of definite integrals.](https://reader038.fdocuments.us/reader038/viewer/2022100505/5a4d1b8e7f8b9ab0599bfe80/html5/thumbnails/6.jpg)
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.
![Page 7: In this section, we will begin to look at notation and how it can be used to represent Riemann sums (rectangle approximations) of definite integrals.](https://reader038.fdocuments.us/reader038/viewer/2022100505/5a4d1b8e7f8b9ab0599bfe80/html5/thumbnails/7.jpg)
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 .
![Page 8: In this section, we will begin to look at notation and how it can be used to represent Riemann sums (rectangle approximations) of definite integrals.](https://reader038.fdocuments.us/reader038/viewer/2022100505/5a4d1b8e7f8b9ab0599bfe80/html5/thumbnails/8.jpg)
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 .
![Page 9: In this section, we will begin to look at notation and how it can be used to represent Riemann sums (rectangle approximations) of definite integrals.](https://reader038.fdocuments.us/reader038/viewer/2022100505/5a4d1b8e7f8b9ab0599bfe80/html5/thumbnails/9.jpg)
Example 5
Evaluate the indicated limit by rewriting it as a definite integral and using the F.T.C.
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Example 6
Evaluate the indicated limit by rewriting it as a definite integral and using the F.T.C.