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ESSENTIAL CALCULUS CH04 Integrals

ESSENTIAL CALCULUS CH04 Integrals

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ESSENTIAL CALCULUS CH04 Integrals

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  1. ESSENTIAL CALCULUSCH04 Integrals

  2. In this Chapter: • 4.1 Areas and Distances • 4.2 The Definite Integral • 4.3 Evaluating Definite Integrals • 4.4 The Fundamental Theorem of Calculus • 4.5 The Substitution Rule Review

  3. Chapter 4, 4.1, P194

  4. Chapter 4, 4.1, P195

  5. Chapter 4, 4.1, P195

  6. Chapter 4, 4.1, P195

  7. Chapter 4, 4.1, P195

  8. Chapter 4, 4.1, P195

  9. Chapter 4, 4.1, P196

  10. Chapter 4, 4.1, P197

  11. Chapter 4, 4.1, P197

  12. Chapter 4, 4.1, P198

  13. Chapter 4, 4.1, P198

  14. Chapter 4, 4.1, P199

  15. 2. DEFINITION The area A of the region S that lies under the graph of the continuous function f is the limit of the sum of the areas of approximating rectangles: A=lim Rn=lim[f(x1)∆x+f(x2) ∆x+‧‧‧+f(xn) ∆x] n→∞ n→∞ Chapter 4, 4.1, P199

  16. Chapter 4, 4.1, P199

  17. This tells us to end with i=n. This tells us to add. This tells us to start with i=m. Chapter 4, 4.1, P199

  18. Chapter 4, 4.1, P199

  19. The area of A of the region S under the graphs of the continuous function f is A=lim[f(x1)∆x+f(x2) ∆x+‧‧‧+f(xn) ∆x] A=lim[f(x0)∆x+f(x1) ∆x+‧‧‧+f(xn-1) ∆x] A=lim[f(x*1)∆x+f(x*2) ∆x+‧‧‧+f(x*n) ∆x] n→∞ n→∞ n→∞ Chapter 4, 4.1, P200

  20. FIGURE 1 A partition of [a,b] with sample points Chapter 4, 4.2, P205

  21. A Riemann sum associated with a partition P and a function f is constructed by evaluating f at the sample points, multiplying by the lengths of the corresponding subintervals, and adding: Chapter 4, 4.2, P205

  22. FIGURE 2 A Riemann sum is the sum of the areas of the rectangles above the x-axis and the negatives of the areas of the rectangles below the x-axis. Chapter 4, 4.2, P206

  23. 2. DEFINITION OF A DEFINITE INTEGRAL If f is a function defined on [a,b] ,the definite integral of f from a to b is the number provided that this limit exists. If it does exist, we say that f is integrable on [a,b] . Chapter 4, 4.2, P206

  24. NOTE 1 The symbol ∫was introduced by Leibniz and is called an integral sign. It is an elongated S and was chosen because an integral is a limit of sums. In the notation is called the integrand and a and b are called the limits of integration; a is the lower limit and b is the upper limit. The symbol dx has no official meaning by itself; is all one symbol. The procedure of calculating an integral is called integration. Chapter 4, 4.2, P206

  25. Chapter 4, 4.2, P206

  26. 3. THEOREM If f is continuous on [a,b], or if f has only a finite number of jump discontinuities, then f is integrable on [a,b]; that is, the definite integral dx exists. Chapter 4, 4.2, P207

  27. 4. THEOREM If f is integrable on [a,b], then where Chapter 4, 4.2, P207

  28. Chapter 4, 4.2, P208

  29. Chapter 4, 4.2, P208

  30. Chapter 4, 4.2, P208

  31. Chapter 4, 4.2, P208

  32. Chapter 4, 4.2, P210

  33. Chapter 4, 4.2, P211

  34. MIDPOINT RULE where and Chapter 4, 4.2, P211

  35. Chapter 4, 4.2, P212

  36. Chapter 4, 4.2, P212

  37. Chapter 4, 4.2, P213

  38. Chapter 4, 4.2, P213

  39. PROPERTIES OF THE INTEGRAL Suppose all the following integrals exist. where c is any constant where c is any constant Chapter 4, 4.2, P213

  40. Chapter 4, 4.2, P214

  41. Chapter 4, 4.2, P214

  42. COMPARISON PROPERTIES OF THE INTEGRAL 6. If f(x)≥0 fpr a≤x≤b. then 7.If f(x)≥g(x) for a≤x≤b, then 8.If m≤f(x) ≤M for a≤x≤b, then Chapter 4, 4.2, P214

  43. Chapter 4, 4.2, P215

  44. 29.The graph of f is shown. Evaluate each integral by interpreting it in terms of areas. • (b) • (c) (d) Chapter 4, 4.3, P217

  45. 30. The graph of g consists of two straight lines and a semicircle. Use it to evaluate each integral. (a) (b) (c) Chapter 4, 4.3, P217

  46. EVALUATION THEOREM If f is continuous on the interval [a,b] , then Where F is any antiderivative of f, that is, F’=f. Chapter 4, 4.3, P218

  47. the notation ∫f(x)dx is traditionally used for an antiderivative of f and is called an indefinite integral. Thus The connection between them is given by the Evaluation Theorem: If f is continuous on [a,b], then Chapter 4, 4.3, P220

  48. ▓You should distinguish carefully between definite and indefinite integrals. A definite integral is a number, whereas an indefinite integral is a function (or family of functions). Chapter 4, 4.3, P220

  49. 1. TABLE OF INDEFINITE INTEGRALS Chapter 4, 4.3, P220

  50. ■ Figure 3 shows the graph of the integrand in Example 5. We know from Section 4.2 that the value of the integral can be interpreted as the sum of the areas labeled with a plus sign minus the area labeled with a minus sign. Chapter 4, 4.3, P221