1. time velocity After 4 seconds, the object has gone 12 feet. Consider an object moving at a constant rate of 3 ft/sec. Since rate. time = distance: If we draw a graph of the velocity, the distance that the object travels is equal to the area under the line. 5.1 Estimating with Finite Sums

2. If the velocity is not constant, we might guess that the distance traveled is still equal to the area under the curve. Example: We could estimate the area under the curve by drawing rectangles touching at their left corners. This is called the Left- hand Rectangular Approximation Method (LRAM). Approximate area: 5.1 Estimating with Finite Sums

3. We could also use a Right-hand Rectangular Approximation Method (RRAM). Approximate area: 5.1 Estimating with Finite Sums

4. Another approach would be to use rectangles that touch at the midpoint. This is the Midpoint Rectangular Approximation Method (MRAM). Approximate area: In this example there are four subintervals. As the number of subintervals increases, so does the accuracy. 5.1 Estimating with Finite Sums

5. Approximate area: width of subinterval With 8 subintervals: The exact answer for this problem is. 5.1 Estimating with Finite Sums

6. Circumscribed rectangles are all above the curve: Inscribed rectangles are all below the curve: 5.1 Estimating with Finite Sums

7. We will be learning how to find the exact area under a curve if we have the equation for the curve. Rectangular approximation methods are still useful for finding the area under a curve if we do not have the equation. 5.1 Estimating with Finite Sums

8. When we find the area under a curve by adding rectangles, the answer is called a Rieman sum. subinterval partition The width of a rectangle is called a subinterval. The entire interval is called the partition. Subintervals do not all have to be the same size. 5.2 Definite Integrals

9. subinterval partition If the partition is denoted by P, then the length of the longest subinterval is called the norm of P and is denoted by. As gets smaller, the approximation for the area gets better. if P is a partition of the interval 5.2 Definite Integrals

10. is called the definite integral of over. If we use subintervals of equal length, then the length of a subinterval is: The definite integral is then given by: 5.2 Definite Integrals

11. Leibnitz introduced a simpler notation for the definite integral: Note that the very small change in x becomes dx. 5.2 Definite Integrals

12. Integration Symbol lower limit of integration upper limit of integration integrand variable of integration (dummy variable) It is called a dummy variable because the answer does not depend on the variable chosen. 5.2 Definite Integrals

13. We have the notation for integration, but we still need to learn how to evaluate the integral. 5.2 Definite Integrals

14. Definition Area Under a Curve (as a Definite Integral) If y = f(x) is non negative and integrable over a closed interval [a,b], then the area under the curve y = f(x) from a to b is the integral of f from a to b. A =

15. time velocity After 4 seconds, the object has gone 12 feet. In section 5.1, we considered an object moving at a constant rate of 3 ft/sec. If we draw a graph of the velocity, the distance that the object travels is equal to the area under the line. 5.2 Definite Integrals

16. If the velocity varies: Distance is the area under the curve 5.2 Definite Integrals

17. Area Where F(x) is the antiderivative of f(x)! 5.2 Definite Integrals

18. What if: We could split the area under the curve into a lot of thin trapezoids, and each trapezoid would behave like the large one in the previous example. It seems reasonable that the distance will equal the area under the curve. 5.2 Definite Integrals

19. The area under the curve We can use anti-derivatives to find the area under a curve! 5.2 Definite Integrals

20. Area from x=0 to x=1 Example:Find the area under the curve from x = 1 to x = 2. Area from x=0 to x=2 Area under the curve from x = 1 to x = 2. 5.2 Definite Integrals

21. Example: Find the area under the curve from x = 1 to x = 2. To do the same problem on the TI-83: fnInt( x 2, x,1,2) 5.2 Definite Integrals

22. Example: Find the area between the x-axis and the curve from to. pos. neg. 5.2 Definite Integrals

23. Page 269 gives rules for working with integrals, the most important of which are: 2. If the upper and lower limits are equal, then the integral is zero. 1. Reversing the limits changes the sign. 3. Constant multiples can be moved outside. 5.3 Definite Integrals and Antiderivatives

24. 1. If the upper and lower limits are equal, then the integral is zero. 2. Reversing the limits changes the sign. 3. Constant multiples can be moved outside. 4. Integrals can be added and subtracted. 5.3 Definite Integrals and Antiderivatives

25. 5. Intervals can be added (or subtracted.) 5.3 Definite Integrals and Antiderivatives

26. 6. 5.3 Definite Integrals and Antiderivatives a b

27. 5.3 Definite Integrals and Antiderivatives Suppose that f and g are continuous functions and that Find 32*5=10 -5 – -3 = -2 -3 + 5 = 203*(-3)- 2 * 6 = -21

28. The average value of a function is the value that would give the same area if the function was a constant: 5.3 Definite Integrals and Antiderivatives

29. For what value(s) in the interval does the function assume the average value? 5.3 Definite Integrals and Antiderivatives

30. The mean value theorem for definite integrals says that for a continuous function, at some point on the interval the actual value will equal the average value. Mean Value Theorem (for definite integrals) If f is continuous on then at some point c in, 5.3 Definite Integrals and Antiderivatives

31. Find the total area bounded by the x-axis and the graph y = x 2 - 3x from [0,4].

32. If you were being sent to a desert island and could take only one equation with you, might well be your choice. Quote from CALCULUS by Ross L. Finney and George B. Thomas, Jr., ©1990. 5.4 Fundamental Theorem of Calculus

33. The Fundamental Theorem of Calculus, Part 1 If f is continuous on, then the function has a derivative at every point in, and 5.4 Fundamental Theorem of Calculus

34. First Fundamental Theorem: 1. Derivative of an integral. 5.4 Fundamental Theorem of Calculus

35. 2. Derivative matches upper limit of integration. First Fundamental Theorem: 1. Derivative of an integral. 5.4 Fundamental Theorem of Calculus

36. 1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant. First Fundamental Theorem: 5.4 Fundamental Theorem of Calculus

37. 1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant. New variable. First Fundamental Theorem: 5.4 Fundamental Theorem of Calculus

38. 1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant. The long way: First Fundamental Theorem: 5.4 Fundamental Theorem of Calculus

39. 1. Derivative of an integral. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant. 5.4 Fundamental Theorem of Calculus

40. The upper limit of integration does not match the derivative, but we could use the chain rule. 5.4 Fundamental Theorem of Calculus

41. The lower limit of integration is not a constant, but the upper limit is. We can change the sign of the integral and reverse the limits. 5.4 Fundamental Theorem of Calculus

42. Neither limit of integration is a constant. It does not matter what constant we use! (Limits are reversed.) (Chain rule is used.) We split the integral into two parts. 5.4 Fundamental Theorem of Calculus

43. The Fundamental Theorem of Calculus, Part 2 If f is continuous at every point of, and if F is any antiderivative of f on, then (Also called the Integral Evaluation Theorem) 5.4 Fundamental Theorem of Calculus

46. #55 Pg 287 f is the differentiable function whose graph is shown in the figure. The position at time t (sec) of a particle moving along a coordinate axis is a.What is the particle’s velocity at t = 3? b.Is the acceleration of the particle at time t = 3 positive or negative? c.What is the particle’s position at t = 3? d.When does the particle pass through the origin? e.Approximately, when is the acceleration zero? f.When does the particle move toward the origin? away? g.On which side of the origin does the particle lie at t = 9? (3,0) y = f(x) (6,6) (7,6.5) s’(3) = f(3) = 0 s’’(3) = f’(3) > 0 0 6; 3 < t < 6 Positive Side, area is larger

47. Using integrals to find area works extremely well as long as we can find the antiderivative of the function. Sometimes, the function is too complicated to find the antiderivative. At other times, we don’t even have a function, but only measurements taken from a real-life object. What we need is an efficient method to estimate area when we can not find the antiderivative. 5.5 Trapezoid Rule

48. Actual area under curve: 5.5 Trapezoid Rule

49. Left-hand rectangular approximation: Approximate area: (too low) 5.5 Trapezoid Rule

50. Approximate area: Right-hand rectangular approximation: (too high) 5.5 Trapezoid Rule

51. Averaging the two: 1.25% error (too high) 5.5 Trapezoid Rule

52. x o =a x 1 x 2 x 3 x n-1 x n = b

53. Averaging right and left rectangles gives us trapezoids: 5.5 Trapezoid Rule

55. (x o,y o ) (x 1,y 1 ) (x 2,y 2 ) (x 3,y 3 ) (x 4,y 4 ) h

56. Trapezoidal Rule: where [a,b] is partitioned into n subintervals of equal length h = (b – a)/n This gives us a better approximation than either left or right rectangles. 5.5 Trapezoid Rule

57. Simpson’s Rule: Uses curved arcs to approximate areas (-h,y o ) (0,y 1 ) (h,y 2 ) L M R hh

58. 5.5 Trapezoid Rule (-h,y o ) (0,y 1 ) (h,y 2 ) L M R

59. 5.5 Trapezoid Rule (-h,y o ) (0,y 1 ) (h,y 2 ) L M R but so

60. Simpson’s Rule: 5.5 Trapezoid Rule where [a,b] is partitioned into an even n subintervals of equal length h = (b – a)/n

61. Example: 5.5 Trapezoid Rule

62. Simpson’s rule can also be interpreted as fitting parabolas to sections of the curve, which is why this example came out exactly. Simpson’s rule will usually give a very good approximation with relatively few subintervals. It is especially useful when we have no equation and the data points are determined experimentally. 5.5 Trapezoid Rule

63. Error Bounds If T and S represent the approximations to given by the Trapezoid Rule and Simpson’s Rule, respectively, then the errors E T and E s satisfy