Homework questions thus far???

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 trapezoidalrule

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

12. Simpson’s Rule Formula Interiorcoefficientsalternate: 4,2,4,2,…,4

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

14. Simpson’s Rule • Uses Parabolas to fit the curve • Where n is even and ∆x = (b - a)/n • S2n=(Tn+ 2Mn)/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 T4.

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 x0(=a),x1,x2,…,xn(=b) be the endpoints of these subintervals and we let x1*, x2*, … xn* be any sample points in these subintervals so xi*lies in the ith subinterval [xi-1,xi]. 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

21. Properties of the Definite Integral

22. 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(x2).

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

40. Example Suppose we define . Then F'(x) = x2 + 2x + 1.

41. Examples:

42. Examples:

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

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

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

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

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

48. Consider the function f(x) = x+1 on the interval [0,3] • 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: