1. 4.2 Area

2. Sigma Notation

3. Summation Examples
Example:
Example:
Example:
Example:

4. More Summation Examples

5. Theorem 4.2 Summation Rules

6. Theorem 4.2 Summation Rules

7. Examples
Example 2 Evaluate the summation
Solution

8. Examples
Example 3 Compute
Solution

9. 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

10. Summation and Limits
Example Find the limit for

11. Continued…

12. Area 2

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

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

15. Continued… The average of the lower and upper approximations is L U L
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(mi) and that of circumscribed rectangle is f(Mi).

18. 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

20. Theorem 4.3 Limits of the Upper and Lower Sums

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

22. Exact Area Using the Limit

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

24. In General - Finding Area Using the Limitb
height x
base
Area =
Or, xi , the i-th right endpoint

25. 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

26. Regular Right-Endpoint Formula

27. Continued

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