1. Area
Section 4.2

2. Summation Sigma is used to denote summation.
MAT SPRING 2007
Summation
Sigma is used to denote summation.
The sum of n terms a1, a2, a3, …, an is expressed as
i is called the ______________________________________ 1 is the ____________________________; n is the ___________________________ . ai is the ith term of the sum.

3. Summation Examples
Example:
Example:
Example:

4. Summation
Example:

5. Summation Rules

6. Summation Rules

7. Examples

8. Area 2

9. Consider the region bounded by the graphs of
The area can be approximated by two sets of rectangles—one set inscribed within the region and the other set circumscribed over the region.
f(x)=x2
Circumscribed rectangles
Upper Sum
f(x)=x2
Inscribed rectangles
Lower Sum
f(x)=x2
The actual area lies between the lower and upper sums.

10. Lower Approximation
Find the sum of the areas of the inscribed rectangles.
f(x)=x2
Inscribed rectangles
Lower Sum

11. Upper Approximation
Find the sum of the areas of the circumscribed rectangles.
f(x)=x2
Circumscribed rectangles
Upper Sum

12. Continued… The actual area lies between the lower and upper sums. L A
Thus, the area bounded by the graphs of is

13. Example
Find the lower and upper approximations of the area of the region lying between the graph of and the x-axis between x = 0 and x = 2. Use 4 rectangles.
1) Lower Sum

14. Example
2) Upper Sum

15. The limit In fact, the exact area can be found by: Area under curve =
**Notice: The smaller the intervals (the greater the number of rectangles), the closer the approximate area is to the actual area.
In fact, the exact area can be found by:
Area under curve =
Lower Sum
Upper Sum

16. As n increases without bound
As you increase the number of rectangles, the approximation tends to become better because the amount of ‘missed area’ decreases.
Check this out:

17. Homework
Section 4.2 page 267 # 1 – 7 odd, 15, 23, 27, 29