1. Lecture 19 Approximating Integrals

2. Integral = “Area”

3. Estimating Using Rectangles

6. Size of Rectangle Base Can Vary

7. ab c f(c) Area of approximating Rectangle = f(c) (b-a)

8. A partition of an Interval [a,b] is just a choice of numbers with for each i = 0.. n-1

9. Examples {1,2,3,4,5,9,12} is a partition of [1,12] {-5,0,20} is a partition of [-5,20]

10. A marking of the partition is just a choice of numbers With for each i = 1.. n

11. Examples {3,7} is a making of the partition {1, 5,12} since 3 is in [1,5] and 7 is in [5,12] {1,2,3,4,5} is a marking of the partition {1,2,3,4,5,6} since 1 is in [1,2], 2 is in [2,3], …

12. Integral Estimates from Partitions If P = Is a partition of [a,b] and is a marking of P then for any function f defined on [a,b] we have an for given by

14. Theorem: As the maximum distance between elements of the partition goes to zero the estimates converge to the actual integral

16. Regular Partitions A partition is regular all of the rectangles it defines have the same Base length. More precisely is regular if is the same for each i. If then n*h = b-a or

17. Regular Estimates for Uses the regular partition of [a,b] into n subintervals and the left end point of each subinterval for the marking (The Trapezoid Rule) is the average of and

18. Example Calculate the left regular estimate for with 5 subdivisions = 2 = = 90

19. Example Calculate the right regular estimate for with 5 subdivisions = 2 = = 170

20. Example Calculate the trapezoid rule estimate for with 5 subdivisions

21. Average of a Function If f(x) is a function defined on an interval [a,b] then the average value of f on [a,b] is

22. The temperatures every two hours from midnight (time t = 0 hrs) to noon is given in the table. Estimate the average temperature over that time interval using the left regular, right regular, and trapezoid estimates. a = 0, b = 12, n = 6 h = 2 The left estimate is =

23. Similarly the right estimate is = 48*2 + 53*2 + 57*2+60*2+62*2+63*2 = 686 The trapezoid rule estimate for the integral is = 666 so the trapezoid rule estimates the average temperature as = 55.5 Remark: The average of the temperatures above is So the average of the temperatures is not the same as the average temperature