1. 11.5 Area 2014

2. After this lesson, you should be able to: Use sigma notation to write and evaluate a sum. Understand the concept of area. Approximate the area of a plane region. Find the area of a plane region using limits.

3. Sigma Notation

4. Summation Examples Example:

5. Example 1 More Summation Examples

6. Summation Rules

8. Example 1 Evaluate the summation Solution Examples

9. Example 2 Compute Solution Examples

10. Example 3 Evaluate the summation for n = 100 and 10000 Solution Note that we change (shift) the upper and lower bound For n = 100For n = 10000 Examples

11. Summation and Limits Example 4 Find the limit for

12. Continued…

13. Review Example : Evaluate the following limit:

14. Area 2 Area of the region bounded by and the lines x=2 and y=0 ?

15. Lower Approximation Using 4 inscribed rectangles of equal width Lower approximation = (sum of the rectangles) 2

16. Using 4 circumscribed rectangles of equal width Upper approximation = (sum of the rectangles) 2 Upper Approximation

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

18. 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

19. 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 ).

20. 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

21. 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.

22. Theorem 4.3 Limits of the Upper and Lower Sums

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

24. 2 length =2 – 0 =2 n = # of rectangles Exact Area Using the Limit

26. Definition of the Area of a Region in the Plane

27. Regular Right-Endpoint Formula RR-EF Example 6 Find the area under the graph of 15 A =

28. Regular Right-Endpoint Formula

29. Continued

30. Regular Right-Endpoint Formula RR-EF Example 7 Find the area bounded by the graph of f(x), the x-axis, the y-axis, and x = 3.

31. Regular Right-Endpoint Formula RR-EF Example 8 Find the area bounded by the graph of f(x), and the x-axis on the given interval

32. Homework Day 1: Section 11.5 pg. 788 1-5 odd, 15-29 odd Day 2: Section 11.5 pg. 788 16-30 even Day 3: Ch. 11 Review pg. 791 3-91 odd Day 4: Ch. 11 Practice Test Ch. 11 Test Monday 5/11

33. HWQ Find the area between the graph of f(x) and the x-axis on the given interval: