1. Homework questions thus far??? Section 4.10? 5.1? 5.2?

2. The Definite Integral Chapters 7.7, 5.2 & 5.3 January 30, 2007

3. Estimating Area vs Exact Area

4. Pictures Riemann sum rectangles, ∆t = 4 and n = 1:

5. Better Approximations Trapezoid Rule uses straight lines

6. Trapezoidal Rule

7. Better Approximations The Trapezoid Rule uses small lines Next highest degree would be parabolas…

8. Simpson’s Rule Mmmm… parabolas… Put a parabola across each pair of subintervals:

9. Simpson’s Rule Mmmm… parabolas… Put a parabola across each pair of subintervals: So n must be even!

10. Simpson’s Rule Formula Like trapezoidal rule

11. Simpson’s Rule Formula Divide by 3 instead of 2

12. Simpson’s Rule Formula Interior coefficients alternate: 4,2,4,2,…,4

13. Simpson’s Rule Formula Second from start and end are both 4

14. Simpson’s Rule Uses Parabolas to fit the curve Where n is even and ∆x = (b - a)/n S 2n =(T n + 2M n )/3

15. Use Simpson’s Rule to Approximate the definite integral with n = 4 g(x) = ln[x]/x on the interval [3,11] Use T 4.

16. Runners: A radar gun was used to record the speed of a runner during the first 5 seconds of a race (see table) Use Simpsons rule to estimate the distance the runner covered during those 5 seconds.

17. Definition of Definite Integral: If f is a continuous function defined for a≤x≤b, we divide the interval [a,b] into n subintervals of equal width ∆x=(b-a)/n. We let x 0 (=a),x 1,x 2,…,x n (=b) be the endpoints of these subintervals and we let x 1 *, x 2 *, … x n * be any sample points in these subintervals so x i * lies in the ith subinterval [x i-1,x i ]. Then the Definite Integral of f from a to b is:

18. Express the limit as a Definite Integral

19. Express the Definite Integral as a limit

20. Properties of the Definite Integral

23. Properties of the Integral 1) 2) = 0 3) for “c” a constant

24. Properties of the Definite Integral Given that:  Evaluate the following:

25. Properties of the Definite Integral Given that:  Evaluate the following:

26. Given the graph of f, find:

27. Evaluate:

28. Integral Defined Functions Let f be continuous. Pick a constant a. Define:

29. Integral Defined Functions Let f be continuous. Pick a constant a. Define: Notes: lower limit a is a constant.

30. Integral Defined Functions Let f be continuous. Pick a constant a. Define: Notes: lower limit a is a constant. Variable is x: describes how far to integrate.

31. Integral Defined Functions Let f be continuous. Pick a constant a. Define: Notes: lower limit a is a constant. Variable is x: describes how far to integrate. t is called a dummy variable; it’s a placeholder

32. Integral Defined Functions Let f be continuous. Pick a constant a. Define: Notes: lower limit a is a constant. Variable is x: describes how far to integrate. t is called a dummy variable; it’s a placeholder F describes how much area is under the curve up to x.

33. Example Let. Let a = 1, and. Estimate F(2) and F(3).

34. Example Let. Let a = 1, and. Estimate F(2) and F(3).

35. Where is increasing and decreasing? is given by the graph below: F is increasing. (adding area) F is decreasing. (Subtracting area)

36. Fundamental Theorem I Derivatives of integrals: Fundamental Theorem of Calculus, Version I: If f is continuous on an interval, and a a number on that interval, then the function F(x) defined by has derivative f(x); that is, F'(x) = f(x).

37. Example Suppose we define.

38. Example Suppose we define. Then F'(x) = cos(x 2 ).

39. Example Suppose we define. Then F'(x) =

40. Example Suppose we define. Then F'(x) = x 2 + 2x + 1.

41. Examples:

43. If f is continuous on [a, b], then the function defined by is continuous on [a, b] and differentiable on (a, b) and Fundamental Theorem of Calculus (Part 1)

44. If f is continuous on [a, b], then the function defined by is continuous on [a, b] and differentiable on (a, b) and Fundamental Theorem of Calculus (Part 1) (Chain Rule)

45. In-class Assignment a. Estimate (by counting the squares) the total area between f(x) and the x- axis. b. Using the given graph, estimate c. Why are your answers in parts (a) and (b) different? 2. 1. Find:

46. First let the bottom bound = 1, if x >1, we calculate the area using the formula for trapezoids: Consider the function f(x) = x+1 on the interval [0,3]

47. Now calculate with bottom bound = 1, and x < 1, : Consider the function f(x) = x+1 on the interval [0,3]

48. So, on [0,3], we have that And F’(x) = x + 1 = f(x) as the theorem claimed! Very Powerful! Every continuous function is the derivative of some other function! Namely: