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4.2 Area. Sigma Notation Summation Examples Example:

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Presentation on theme: "4.2 Area. Sigma Notation Summation Examples Example:"— Presentation transcript:

1 4.2 Area

2 Sigma Notation

3 Summation Examples Example:

4 Example 1 More Summation Examples

5 Theorem 4.2 Summation Rules


7 Example 2 Evaluate the summation Solution Examples

8 Example 3 Compute Solution Examples

9 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

10 Summation and Limits Example 5 Find the limit for

11 Continued…

12 Area 2

13 Lower Approximation Using 4 inscribed rectangles of equal width Lower approximation = (sum of the rectangles) 2 The total number of inscribed rectangles

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

15 Continued… LU LAU A The average of the lower and upper approximations is A is approximately

16 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

17 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 ).

18 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


20 Theorem 4.3 Limits of the Upper and Lower Sums

21 2 length =2 – 0 =2 n = # of rectangles Exact Area Using the Limit


23 Definition of the Area of a Region in the Plane

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

25 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 =

26 Regular Right-Endpoint Formula

27 Continued

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

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