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4.2 Area

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Sigma Notation

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Summation Examples Example:

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Example 1 More Summation Examples

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Theorem 4.2 Summation Rules

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Example 2 Evaluate the summation Solution Examples

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Example 3 Compute Solution Examples

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Example 4 Evaluate the summation for n = 100 and Solution Note that we change (shift) the upper and lower bound For n = 100For n = Examples

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Summation and Limits Example 5 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 Lower approximation = (sum of the rectangles) 2 The total number of inscribed rectangles

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Using 4 circumscribed rectangles of equal width Upper approximation = (sum of the rectangles) 2 Upper Approximation The total number of circumscribed rectangles

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Continued… LU LAU A The average of the lower and upper approximations is 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(m i ) and that of circumscribed rectangle is f(M i ).

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Upper and Lower Sums The i-th regional area A i 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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2 length =2 – 0 =2 n = # of rectangles Exact Area Using the Limit

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Definition of the Area of a Region in the Plane

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ab Area = heightxbase In General - Finding Area Using the Limit Or, x i, the i -th right endpoint

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Regular Right-Endpoint Formula RR-EF intervals are regular in length squaring from right endpt of rect. Example 6 Find the area under the graph of 15 a = 1 b = 5 A =

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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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