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MAT 1221 Survey of Calculus

MAT 1221 Survey of Calculus. Section 6.4 Area and the Fundamental Theorem of Calculus. http://myhome.spu.edu/lauw. Quiz. 8 minutes. Bonus Event (5/9). Your feedback is very helpful to the speaker. Bonus Event. Friday 5/23; 5:10- 5:40 (Hedging), 5:45-6:15 (Genetics Inbreeding Problem)

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MAT 1221 Survey of Calculus

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  1. MAT 1221Survey of Calculus Section 6.4 Area and the Fundamental Theorem of Calculus http://myhome.spu.edu/lauw

  2. Quiz • 8 minutes

  3. Bonus Event (5/9) • Your feedback is very helpful to the speaker.

  4. Bonus Event • Friday 5/23; 5:10- 5:40 (Hedging), 5:45-6:15 (Genetics Inbreeding Problem) • You will get 2 points for the first presentation and 1 point for the second one. • If you participated in the 5/9 event, you will get an additional bonus point for attending both presentations.

  5. Major Themes in Calculus

  6. Major Themes in Calculus • We do not like to use the definition • Develop techniques to deal with different functions

  7. Major Themes in Calculus

  8. Major Themes in Calculus • We do not like to use the definition • Develop techniques to deal with different functions

  9. Preview • Look at the definition of the definite integral on • Look at its relationship with the area between the graph and the -axis on • Properties of Definite Integrals • The Substitution Rule for Definite Integrals

  10. Key • Pay attention to the overall ideas • Pay less attention to the details – We are going to use a formula to compute the definite integrals, not limits.

  11. Example 0

  12. Example 0 Use left hand end points to get an estimation

  13. Example 0 Use right hand end points to get an estimation

  14. Example 0 Observation: What happen to the estimation if we increase the number of subintervals?

  15. In General ith subinterval sample point

  16. In General Suppose is a continuous function defined on , we divide the interval into n subintervals of equal width The area of the rectangle is

  17. In General sample point subinterval

  18. In General Sum of the area of the rectangles is Riemann Sum

  19. In General Sum of the area of the rectangles is Sigma Notation for summation

  20. In General Sum of the area of the rectangles is Final value (upper limit) Initial value (lower limit) Index

  21. In General Sum of the area of the rectangles is As we increase , we get better and better estimations.

  22. Definition The Definite Integral of from to

  23. Definition The Definite Integral of from to upper limit lower limit integrand

  24. Definition The Definite Integral of from to Integration : Process of computing integrals

  25. Remarks • We are not going to use this limit definition to compute definite integrals. • We are going to use antiderivative (indefinite integral) to compute definite integrals.

  26. Area and Indefinite Integrals If on , then from to .

  27. Area and Indefinite Integrals Otherwise, the definite integral may not have obvious geometric meaning.

  28. Example 1 Compute by interpreting it in terms of area.

  29. Example 1 We are going to use this example to verify our next formula.

  30. Fundamental Theorem of Calculus Suppose is continuous on and is any antiderivative of . Then

  31. Remarks • To simplify the computations, we always use the antiderivative with C=0.

  32. Remarks • To simplify the computations, we always use the antiderivative with C=0. • We will use the following notation to stand for F(b)-F(a):

  33. FTC Suppose is continuous on and is any antiderivative of . Then

  34. Example 2

  35. Example 3

  36. Example 4

  37. The Substitution Rule for Definite Integrals • For complicated integrands, we use a version of the substitution rule.

  38. The Substitution Rule for Definite Integrals • The procedures for indefinite and definite integrals are similar but different. • We need to change the upper and lower limits when using a substitution. • Do not change back to the original variable.

  39. The Substitution Rule for Definite Integrals

  40. The Substitution Rule for Definite Integrals

  41. Example 5

  42. Example 6

  43. Physical Meanings of Definite Integrals • We will not have time to discuss the exact physical meanings. • Basic Idea: The definite integral of rate of change is the net change.

  44. Example 7 (HW 18) • A company purchases a new machine for which the rate of depreciation can be modeled by the equation below, where  is the value of the machine after  years. • Find the total loss of value of the machine over the first 4 years.

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