Math-412-Emilyn-Diosana_DEFINITE-INTEGRALS.pptx

this a PowerPoint presentation about definite integrals, This presentation is for subject math 402 the analytic geometry.

1. DEFINITE INTEGRALS Presented by: Emilyn N. Diosana Math 412 – Mathematical Analysis

2. Summation Notation and Riemann Sum Definition of Definite Integrals Properties of Definite Integrals

3. Find the sum of the first 25 even natural numbers.

4. SUMMATION NOTATION Summation notation (sigma notation) allows us to write a long sum in a very concise manner.   Sigma Symbol It tells us that we are summing something.

5. SUMMATION NOTATION

6. How to write Sigma Notation? Find the general term of the terms of the sum. Select some alphabet (preferably a lowercase letter) to be the index. The usual letters we chose for the index of summation is “ i ” or “k”. Observe the sequence and decide the first value and last value of the index. Finally, use the sigma symbol to write the sigma notation.

7. Write the sigma notation for 1 + 2 + 3 + 4 + 5.   Notice that the general term is just i and that there are 5 terms, so we would write

8. Write the sigma notation for 16 + 25 + 36 + 49 + … + 100.    

9. Express each of the following in sigma notation:      

10. Substitute every value of the index in the general term. Place a plus symbol between all such terms obtained from the last step . How to Expand Summation Notation?

11. Find the sum of the first 25 even natural numbers.      

12. Example 2: Evaluate        

13. Evaluate the following:        

14. RIEMANN SUM

15. Riemann Sum A Riemann Sum is an approximation of the area under a curve by dividing it into multiple simple shapes (like rectangles or trapezoids). It is the sum of the areas of the rectangles that cover the region.

16. Riemann Sum The Riemann sum formula is     where = the area under the curve on the interval being evaluated = the height of each rectangle = the width of each rectangle, then is the starting point ; is the endpoint is the number of pieces in which the interval is subdivided      

17. How can we approximate the area of a region under a curve? Left Riemann Sum Right Riemann Sum Midpoint Riemann Sum

18. Left Riemann Sum It is obtained by adding the areas of rectangles whose upper left corners touch the curve.

19. Left Riemann Sum Suppose we want to approximate the area of the region under the curve between and .  

20. Left Riemann Sum Suppose we want to approximate the area of the region under the curve between and .   Rectangle Height Width Area A 4 1 4 B 5 1 5 C 4 1 4 Total 13 Area approximation using left Riemann sum:  

21. Right Riemann Sum It is obtained by adding the areas of rectangles whose upper right corners touch the curve.

22. Right Riemann Sum Suppose we want to approximate the area of the region under the curve between and .   Rectangle Height Width Area A 5 1 5 B 4 1 4 C 1 1 1 TOTAL 10 Area approximation using right Riemann sum:  

23. Midpoint Riemann Sum It is obtained by adding the areas of rectangles whose top middle parts touch the curve.

24. Midpoint Riemann Sum Suppose we want to approximate the area of the region under the curve between and .   Rectangle Height Width Area A 4.75 1 4.75 B 4.75 1 4.75 C 2.75 1 2.75 TOTAL 12.25 Area approximation using Midpoint Riemann sum:  

25. Approximate the area of the region under between and using five rectangles and the left, right, and midpoint Riemann sums.  

26. Rectangle Height Width Area A 1 1 1 B 5 1 5 C 7 1 7 D 7 1 7 E 5 1 5 Total 25 Area approximation using left Riemann sum:   A B C D E Left Riemann Sum

27. Rectangle Height Width Area A 5 1 5 B 7 1 7 C 7 1 7 D 5 1 5 E 1 1 1 Total 25 Area approximation using right Riemann sum:   A B C D E Right Riemann Sum

28. Rectangle Height Width Area A 3.25 1 3.25 B 6.25 1 6.25 C 7.25 1 7.25 D 6.25 1 6.25 E 3.25 1 3.25 Total 26.25 Area approximation using midpoint Riemann sum:   A B C D E Midpoint Riemann Sum

29. Note! Riemann sums sometimes overestimate and other times underestimate .

30. Is this Riemann sum an overestimation or underestimation of the actual area? T he Riemann sum is an  overestimation  of the actual area.

31. Approximations without the aid of graphs.

32. Riemann Sum The Riemann sum formula is     where = the area under the curve on the interval being evaluated = the height of each rectangle = the width of each rectangle, then is the starting point ; is the endpoint is the number of pieces in which the interval is subdivided      

33. Now imagine we're asked to approximate the area between the  x-axis and the curve of     on the interval [0,2] with four equal subdivisions.   ;     x 0.5 1 1.5 2 F(x) 1 1.25 2 3.25         Left-endpoints Riemann Sum:    

34. Now imagine we're asked to approximate the area between the  x-axis and the curve of     on the interval [0,2] with four equal subdivisions.   ;     x 0.5 1 1.5 2 F(x) 1 1.25 2 3.25 Left Riemann Sum:        

35. Now imagine we're asked to approximate the area between the  x-axis and the curve of     on the interval [0,2] with four equal subdivisions.   ;     x 0.5 1 1.5 2 F(x) 1.25 2 3.25 5       Right-endpoints Riemann Sum:      

36. Now imagine we're asked to approximate the area between the  x-axis and the curve of     on the interval [0,2] with four equal subdivisions.   ;     x 0.5 1 1.5 2 F(x) 1.25 2 3.25 5 Right Riemann Sum:        

37. Now imagine we're asked to approximate the area between the  x-axis and the curve of     on the interval [0,2] with four equal subdivisions.   ;     x 0.5 1 1.5 2       Midpoint Riemann Sum:     0.25 0.75 1.25 1.75 F(x) 1.0625 1.5625 2.5625 4.0625

38. Now imagine we're asked to approximate the area between the  x-axis and the curve of     on the interval [0,2] with four equal subdivisions.   ;     Midpoint Riemann Sum:         x 0.5 1 1.5 2 0.25 0.75 1.25 1.75 F(x) 1.0625 1.5625 2.5625 4.0625

39. Now imagine we're asked to approximate the area between the  x-axis and the curve of     on the interval [0,4] with four equal subdivisions.  

40. Now imagine we're asked to approximate the area between the  x-axis and the curve of    on the interval [0,4] with four equal subdivisions.   ;     x 1 2 3 4 F(x) 1 8 27   Left-endpoints Riemann Sum:          

41. Now imagine we're asked to approximate the area between the  x-axis and the curve of    on the interval [0,4] with four equal subdivisions.   x 1 2 3 4 F(x) 1 8 27 Left Riemann Sum:         ;    

42. Now imagine we're asked to approximate the area between the  x-axis and the curve of    on the interval [0,4] with four equal subdivisions.   ;     x 1 2 3 4 F(x) 1 8 27 64         Right-endpoints Riemann Sum:    

43. Now imagine we're asked to approximate the area between the  x-axis and the curve of    on the interval [0,4] with four equal subdivisions.   x 1 2 3 4 F(x) 1 8 27 64 ;     Right Riemann Sum:        

44. x 1 2 3 4 Midpoint Riemann Sum:   0.5 1.5 2.5 3.5 F(x) 0.125 3.375 15.625 42.875 Now imagine we're asked to approximate the area between the  x-axis and the curve of    on the interval [0,4] with four equal subdivisions.   ;            

45. Midpoint Riemann Sum:         Now imagine we're asked to approximate the area between the  x-axis and the curve of    on the interval [0,4] with four equal subdivisions.   ;     x 1 2 3 4 0.5 1.5 2.5 3.5 F(x) 0.125 3.375 15.625 42.875

46. DEFINITE INTEGRALS

47. Definite Integral The limit of the Riemann sum is equal to the definite integral: w here and are the boundaries of the region.  

48. Definite Integral Let be a continuous function on the interval The area of the region between the curve and the from to is given by the definite integral: w here is called the lower limit, and is called the upper limit.  

49. Definite Integrals Express the following limit as an integral on the interval [0,3].  

50. Properties of Definite Integrals

51. Sum/Difference   If f and g are integrable on the on the closed interval [a, b] , then the sum or difference property applies,

52. Sum/Difference   Given: and   Evaluate:          

53. Sum/Difference If and , evaluate the following:      

54. Constant Multiple   If c is a constant, then it can be taken outside of the integral.

55. Constant Multiple   Evaluate given .        

56. Reverse Interval   The sign of the value of the definite integral changes when the limits of integration are interchanged.

57. Reverse Interval   Evaluate , given that .        

58. Zero-length Interval   The definite integral is zero if the integration limits are the same.

59. Additive Interval   If f is integrable on [ a,b ] and on subintervals [ a,c ] and [ c,b ] , where c is a number between a and b , then

60. Additive Intervals If f(x) is a function and and , evaluate .          

61. Additive Intervals If and , evaluate .        

62. Let’s Practice! Given , , and , compute the following definite integrals.     (Reverse Interval)   (Zero-length Interval)   (Constant Multiple)   (Constant Multiple and Sum or Difference) = =  

63. Let’s Practice! Given , , and , compute the following definite integrals.    

64. Let’s Practice! Given , , and , compute the following definite integrals.     (Additive Interval)   (Constant Multiple)  

65. Let’s Practice! Given , , and , compute the following definite integrals.     (Sum or Difference)   (Sum or Difference and Constant Multiple )  

66. https:// www.khanacademy.org/math/ap-calculus-ab/ab-integration-new/ab-6-6/a/definite-integrals-properties-review https://www.cuemath.com/algebra/sigma-notation/? fbclid=IwAR2HJIYkwMA57QBR5qBr3kx1xJ8SBmiT-Yd8AZ0RpsWkjbQ72MqwO1a10Pk https:// www.khanacademy.org/math/ap-calculus-ab/ab-integration-new/ab-6-2/a/left-and-right-riemann-sums?fbclid=IwAR12h4TYSSZIDWmUQhO_HQU1JO3kug9ssdBh3O5ByZPBFF566XdsUp_DbQw https:// www.khanacademy.org/math/ap-calculus-ab/ab-integration-new/ab-6-3/a/review-summation-notation?fbclid=IwAR05acHcGe72U6C9iUaRBHnPt9ggAsOq7r1yU9yrP8rJezRQJoSTkyAdcl0 https://chitowntutoring.com/properties-of-definite-integrals/? fbclid=IwAR32IgxDbuzeZN6hFUpDs2p3nEhl5UFw_7T9kkZLuyvsBarFbuYVaRyqwCM REFERENCES: