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4.2 Area

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Sigma Notation

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Summation Examples Example: Example: Example: Example:

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**More Summation Examples**

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**Theorem 4.2 Summation Rules**

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**Theorem 4.2 Summation Rules**

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Examples Example 2 Evaluate the summation Solution

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Examples Example 3 Compute Solution

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**Examples Example 4 Evaluate the summation for n = 100 and 10000**

Note that we change (shift) the upper and lower bound Solution For n = 100 For n = 10000

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Summation and Limits Example Find the limit for

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Continued…

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Area 2

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**Lower Approximation Using 4 inscribed rectangles of equal width**

The total number of inscribed rectangles 2 Lower approximation = (sum of the rectangles)

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**Upper Approximation Using 4 circumscribed rectangles of equal width**

The total number of circumscribed rectangles 2 Upper approximation = (sum of the rectangles)

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**Continued… The average of the lower and upper approximations is L U L**

A is approximately

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Upper and Lower Sums The procedure we just used can be generalized to the methodology to calculate the area of a plane region. We begin with subdividing the interval [a, b] into n subintervals, each of equal width x = (b – a)/n. The endpoints of the intervals are

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Upper and Lower Sums Because the function f(x) is continuous, the Extreme Value Theorem guarantees the existence of a minimum and a maximum value of f(x) in each subinterval. We know that the height of the i-th inscribed rectangle is f(mi) and that of circumscribed rectangle is f(Mi).

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Upper and Lower Sums The i-th regional area Ai is bounded by the inscribed and circumscribed rectangles. We know that the relationship among the Lower Sum, area of the region, and the Upper Sum is

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**Theorem 4.3 Limits of the Upper and Lower Sums**

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**Exact Area Using the Limit**

length = 2 – 0 = 2 n = # of rectangles 2

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**Exact Area Using the Limit**

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**Definition of the Area of a Region in the Plane**

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**In General - Finding Area Using the Limit**

b height x base Area = Or, xi , the i-th right endpoint

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**Regular Right-Endpoint Formula**

squaring from right endpt of rect. RR-EF intervals are regular in length Example 6 Find the area under the graph of A = 1 5 a = 1 b = 5

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**Regular Right-Endpoint Formula**

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Continued

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Homework Pg , 7, 11, 15, 21, 31, 33, 41, odd, 39, 43

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1 Example 1 Estimate by the six Rectangle Rules using the regular partition P of the interval [0,1] into 4 subintervals. Solution This definite integral.

1 Example 1 Estimate by the six Rectangle Rules using the regular partition P of the interval [0,1] into 4 subintervals. Solution This definite integral.

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