Understanding Sigma Notation and Approximating Areas Using Limits

1. CHAPTER 4SECTION 4.2AREA

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

9. 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.

11. Approximate area: Left endpoint approximation: (too low)

12. Approximate area: Right endpoint approximation: (too high) Averaging the right and left endpoint approximations: (closer to the actual value)

13. Approximating definite integrals:different choices for the sample points • If xi* is chosen to be the leftendpoint of the interval, then xi* = xi-1 and we have • If xi* is chosen to be the right endpoint of the interval, then xi* = xi and we have • Lnand Rn are called the left endpoint approximation and right endpoint approximation , respectively.

14. Approximate area: Can also apply midpoint approximation: choose the midpoint of the subinterval as the sample point. The midpoint rule gives a closer approximation than the trapezoidal rule, but in the opposite direction.

15. Midpoint rule

16. Approximating the Area of a Plane Region a. b. y y f(x) = -x2 + 5 f(x) = -x2 + 5 5 5 4 4 3 3 2 2 1 1 x x 2/5 4/5 6/5 8/5 2 2/5 4/5 6/5 8/5 2 To approximate the area under each curve, you must sum the area of each rectangle. See next slide

17. The right endpoints, Mi, of the intervals are 2i/5, where i = 1, 2, 3, 4, 5. The width of each rectangle is 2/5 and the height of each rectangle can be obtained by evaluating f at the right endpoint of each interval. [0, 2/5], [2/5, 4/5], [4/5, 6/5], [6/5, 8/5], [8/5, 10/5] Evaluate f(x) at the right endpoints of each of these intervals. The sum of the area of the five rectangles is HeightWidth Because each of the five rectangles lies inside the parabolic region, you can conclude that the area of the parabolic region is greater than 6.48.

18. Approximating the Area of a Plane Region for b (con’t) b. The left endpoints of the five intervals are 2/5(i _1), where i = 1, 2, 3, 4, 5. The width of each rectangle is 2/5, and the height of each rectangle can be found by evaluating f at the left endpoint of each interval. HeightWidth Because the parabolic region lies within the union of the five rectangular region, that the area of the parabolic region is less than 8.08. 6.48 < Area of region < 8.08

19. ON CALCULATOR

20. In general for the upper sum S(n) and Lower sum s(n), you use the following for curves f(x) bound between x=a and x=b.

21. Finding Upper and Lower Sums for a Region Find the upper and lower sums for the region bounded by the graph of f(x) = x2 and the x-axis between x = 0 and x = 2 Solution Begin by partitioning the interval [0, 2] into n subintervals, each of length B. A. f(x) = x2 f(x) = x2 4 4 3 3 2 2 1 1 1 2 1 2

22. Left endpointsRight endpoints

24. Limit of the Lower and Upper Sums

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

26. 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.

27. 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.

28. 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.

29. Theorem 4.3 Limits of the Lower and Upper Sums

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

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

32. 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.

33. 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.

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

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

36. …continued Foil & Simplify

37. 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

38. Practice with Limits Multiply out Separate