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![Page 1: In this section, we will investigate how to estimate the value of a definite integral when geometry fails us. We will also construct the formal definition.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649f435503460f94c63cc2/html5/thumbnails/1.jpg)
In this section, we will investigate how to estimate the value of a definite integral when geometry fails us. We will also construct the formal definition of the definite integral.
Section 5.6 Approximating Sums
Formal Definition of the Definite Integral
![Page 2: In this section, we will investigate how to estimate the value of a definite integral when geometry fails us. We will also construct the formal definition.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649f435503460f94c63cc2/html5/thumbnails/2.jpg)
Idea
Consider the function f(x) = sin(x) + 2 and its integral over the interval [1, 3].
![Page 3: In this section, we will investigate how to estimate the value of a definite integral when geometry fails us. We will also construct the formal definition.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649f435503460f94c63cc2/html5/thumbnails/3.jpg)
Idea
Consider the function f(x) = sin(x) + 2 and its integral over the interval [1, 3].
The area of this region is not able to be found using basic geometric formulas like we did in section 5.1
![Page 4: In this section, we will investigate how to estimate the value of a definite integral when geometry fails us. We will also construct the formal definition.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649f435503460f94c63cc2/html5/thumbnails/4.jpg)
Idea
Consider the function f(x) = sin(x) + 2 and its integral over the interval [1, 3].
The area of this region is not able to be found using basic geometric formulas like we did in section 5.1
Instead, we estimate the value of the area by using some number of rectangles (left, right, or midpoint) or trapezoids.
![Page 5: In this section, we will investigate how to estimate the value of a definite integral when geometry fails us. We will also construct the formal definition.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649f435503460f94c63cc2/html5/thumbnails/5.jpg)
Rectangle Process
1. Divide the interval [a, b] into n subintervals each of width
2. For Ln or Rn, use the appropriate point from each subinterval (called xi). The height of each rectangle is and its area is .
3. For Mn, use the midpoint of each subinterval (called mi). The height of each rectangle is and its area is .
4. Add together the area of the n rectangles to get the desired approximation.
![Page 6: In this section, we will investigate how to estimate the value of a definite integral when geometry fails us. We will also construct the formal definition.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649f435503460f94c63cc2/html5/thumbnails/6.jpg)
Trapezoid Process
1. Divide the interval [a, b] into n subintervals each of width
2. In the ith subinterval, use both of the endpoints of that subinterval, xi and xi+1. The area of this trapezoid is .
3. Add together the area of the n trapezoids to get the desired approximation.
![Page 7: In this section, we will investigate how to estimate the value of a definite integral when geometry fails us. We will also construct the formal definition.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649f435503460f94c63cc2/html5/thumbnails/7.jpg)
4 Left Rectangles
Consider the function f(x) = sin(x) + 2 and its integral over the interval [1, 3].
We use the endpoints:
![Page 8: In this section, we will investigate how to estimate the value of a definite integral when geometry fails us. We will also construct the formal definition.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649f435503460f94c63cc2/html5/thumbnails/8.jpg)
4 Right Rectangles
Consider the function f(x) = sin(x) + 2 and its integral over the interval [1, 3].
We use the endpoints:
![Page 9: In this section, we will investigate how to estimate the value of a definite integral when geometry fails us. We will also construct the formal definition.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649f435503460f94c63cc2/html5/thumbnails/9.jpg)
4 Midpoint Rectangles
Consider the function f(x) = sin(x) + 2 and its integral over the interval [1, 3].
We use the midpoints:
![Page 10: In this section, we will investigate how to estimate the value of a definite integral when geometry fails us. We will also construct the formal definition.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649f435503460f94c63cc2/html5/thumbnails/10.jpg)
4 Trapezoids
Consider the function f(x) = sin(x) + 2 and its integral over the interval [1, 3].
We use all the endpoints:
![Page 11: In this section, we will investigate how to estimate the value of a definite integral when geometry fails us. We will also construct the formal definition.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649f435503460f94c63cc2/html5/thumbnails/11.jpg)
Example 1
Find the L4, R4, M4, and T4 approximation for
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Question?
What can we do to get more accurate approximations?
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Question?
What can we do to get more accurate approximations?
Use midpoint rectangles.
They are, in general, more accurate than left or right rectangles or
trapezoids.
![Page 14: In this section, we will investigate how to estimate the value of a definite integral when geometry fails us. We will also construct the formal definition.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649f435503460f94c63cc2/html5/thumbnails/14.jpg)
Question?
What can we do to get more accurate approximations?
Use midpoint rectangles.
They are, in general, more accurate than left or right rectangles or
trapezoids.
Use a larger value of n The more shapes used, the more accurate the approximation
![Page 15: In this section, we will investigate how to estimate the value of a definite integral when geometry fails us. We will also construct the formal definition.](https://reader035.fdocuments.us/reader035/viewer/2022070411/56649f435503460f94c63cc2/html5/thumbnails/15.jpg)
Definition
The definite integral of f from x = a to x = b is formally defined by: