1. Area Use sigma notation to write and evaluate a sum
Section 5.2
Area
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.

2.  SIGMA Notation The sum of n terms a1 , a2 , a3 ,…, an is written as
Where i is the index of summation, ai is the ith term of the sum, and the upper and lower bounds of summation are n and 1.
Note: i does not have to be 1. Any integer less than or equal to the upper bound is legitimate.

3. Examples
Note: The same sum can be represented in different ways using sigma notation.

4. Examples (cont.)

5. Variables
Although any variable can be used as the index of summation, i, j, and k are most often used.

6. Summation Properties

7. Summation Formulas
Theorem

8. Try this one….
For n = 10, 100, 1000, 10,000
Evaluate:

9. For n = 10, 100, 1000, 10,000
Evaluate:
So, …. n 10
100
1000
10,000
What do you think this limit is? .5

10. Time out for a construction! Area Now on to
What’s summation got to do with it?
Time out for a
construction!
This construction is brought to you by the Greek Mathematician Archimedes!

11. Back to our favorite function… y = x2
What’s the area under this curve from x = 0 to x = 2?

12. y = x2 What’s the area under the curve from
x = 0 to x = 2? Let’s approximate it with rectangles…
y = x2
B – A n
x =
n = x = _____
Inscribed rectangles
Ht. of rect. 1?________
Ht. of rect. 2? ________
Lower sum =

13. y = x2 What’s the area under the curve from
x = 0 to x = 2? Let’s approximate it with rectangles…
y = x2
B – A n
x =
n = x = _____
Circumscribed rectangles Ht. of rect. 1?________
Ht. of rect. 2? ________ Upper sum =

14. y = x2 What’s the area under the curve from
x = 0 to x = 2? Let’s approximate it with rectangles…
y = x2
B – A n
x=
n = 4 x = _____ Inscribed rectangles Lower sum =
n = x = _____ Circumscribed rectangles Upper sum =

15. So, what is the area under y = x2 from x = 0 to x = 2?
Read and take notes on section 5.2 (p. 295 – 303)
Use sigma notation and limits to find this area after reading 5.2! Be able to explain and discuss this tomorrow!
Do p. 303 # multiple of 3’s from 3 to 45 (i.e. 3, 6, 9, 12, …, 39, 42, 45)

16. Limits, Sigmas and all that Jazz….
In other words…..What have we learned from Archimedes?
We can find the area under a curve with the help of sigma notation and limits. We can also approximate area with rectangles.
Lower sums are found by summing up the areas of inscribed rectangles where n is the number of rectangles, x is the width of each rectangle, and f(mi) is the height of each inscribed rectangle.
Upper sums are found by summing up the areas of circumscribed rectangles where n is the number of rectangles, x is the width of each rectangle, and f(Mi) is the height of each circumscribed rectangle.
As the number of rectangles approach infinity, the lower sums = the upper sums.

17. Left vs. Right Endpoints
Find ∆x which is the width of each partition on the interval [A, B]
The left endpoints can be found using the following formula if i = 1. Why? A + (i -1) x
The right endpoints can be found using the following formula if i = 1. Why? A + (i) x
B – A n
x=

18. Increasing Functions: Upper vs. lower sums
Lower Sum – Inscribed Rectangles
In each rectangle which endpoint is used in the function to determine the height of the rectangle?

19. Increasing Functions: Upper vs. Lower sums
Upper Sum – Circumscribed Rectangles
In each rectangle which endpoint is used in the function to determine the height of the rectangle?

20. Is this true for Decreasing functions also?
Lower Sum – Inscribed Rectangles
In each rectangle which endpoint is used in the function to determine the height of the rectangle?

21. Decreasing Functions: Upper vs. Lower sums
Upper Sum – Circumscribed Rectangles
In each rectangle which endpoint is used in the function to determine the height of the rectangle?

22. Uh Oh...
Now what happens if the function doesn’t stay strictly increasing or decreasing and I want to find lower or upper sums?

23. Right....there’s LIMITS!
The World is HAPPY!

24. Limits to the Rescue! Who would have guessed?
where ∆x = (B-A)/n
Homework: (yep that’s right....time to practice)
P. 304 # 46, 49, 52, ...., 73,