1. 4.2 Area Under a Curve

2. Sigma (summation) notation REVIEW
In this case k is the index of summation
The lower and upper bounds of summation are 1 and 5
In this case i is the index of summation
The lower and upper bounds of summation are 1 and 6

3. Sigma notation

4. Sigma Summation Notation

5. Practice with Summation Notation
= 3080

6. Practice with Summation Notation
Numerical Problems can be done with the TI83+/84 as was done in PreCalc Algebra
Sum is in LIST, MATH
Seq is on LIST, OPS

7. TI-89 Calculator Steps Try: 𝑖=1 25 2 𝑖 2 −4 Answer: 10,950
Hit F3, go down to #4 sum
Enter in the equation
Comma x
Comma lower bound
Comma upper bound
End the parenthesis
Try: 𝑖= 𝑖 2 −4
Answer: 10,950

8. Area Under a Curve by Limit Definition
The area under a curve can be approximated by the sum of rectangles. The figure on the left shows inscribed rectangles while the figure on the right shows circumscribed rectangles
This gives the lower sum.
This gives the upper sum.

9. Area Under a Curve by Limit Definition
If the width of each of n rectangles is x, and the height is the minimum value of f in the rectangle, f(Mi), then the area is the limit of the area of the rectangles as n 
This gives the lower sum.

10. Area under a curve by limit definition
If the width of each of n rectangles is x, and the height is the maximum value of f in the rectangle, f(mi), then the area is the limit of the area of the rectangles as n 
This gives the upper sum.

11. Area under a curve by limit definition
The limit as n  of the Upper Sum =
The limit as n  of the Lower Sum =
The area under the curve between x = a and x = b.

12. Area under a curve by limit definition
ci is any point in the interval
Area =
f is continuous on [a,b]
f(ci) a b ci

13. Visualization
f(ci) ci
Width = Δx
ith interval

14. Example: Area under a curve by limit definition
Find the area of the region bounded by the graph f(x) = 2x – x3 , the x-axis, and the vertical lines x = 0 and x = 1, as shown in the figure.

15. Area under a curve by limit definition
Why is right, endpoint i/n?
Suppose the interval from 0 to 1 is divided into 10 subintervals, the endpoint of the first one is 1/10, endpoint of the second one is 2/10 … so the right endpoint of the ith is i/10.

16. Visualization again
f(ci)
ci = i/n
Width = Δx=
ith interval

17. Sum of all the rectangles
Find the area of the region bounded by the graph f(x) = 2x – x3 on [0, 1]
Sum of all the rectangles
Right endpoint
Sub for x in f(x)
Use rules of summation

18. …continued
Foil & Simplify

19. The area of the region bounded by the graph f(x) = 2x – x3 , the x-axis, and the vertical lines x = 0 and x = 1, as shown in the figure = .75
0.75

20. Practice with Limits
Multiply out
Separate

21. 4.2 Area
Please read the text and go over the examples in the text. Carefully do the assignment making sure you can work out the summation notation. This is an important section though we only have one day to spend on it.
Assignment: p #1-65 odd