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Integral Calculus

Integral Calculus. Problems. 2-Variable Function with a Maximum. z = f(x,y). Sequence Problem Solving. ?? 49 36 18 8 2 5 11 17 23 ? 3 3 5 4 4 3 5 ? . 2-Variable Function with both Maxima and Minima. z = f(x,y). 2-Variable Function with a Saddle Point. z = f(x,y).

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Integral Calculus

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  1. Integral Calculus Problems rd

  2. 2-Variable Function with a Maximum z = f(x,y) rd

  3. Sequence Problem Solving • ?? 49 36 18 8 • 2 5 11 17 23 ? • 3 3 5 4 4 3 5 ?

  4. 2-Variable Function with both Maxima and Minima z = f(x,y) rd

  5. 2-Variable Function with a Saddle Point z = f(x,y) rd

  6. Natural Logarithms With x/h = n tending to infinity yielding e rd

  7. Integration • Integral Calculus • Stochastic (Probability) Models • Differential Equations • Dynamic Models rd

  8. Integration If F(x) is a function whose derivative F’(x) = f(x), then F(x) is called the integral of f(x) For example, F(x) = x3 is an integral of f(x) = 3x2 Note also that G(x) = x3 + 5 and H(x) = x3 – 6 are also integrals of f(x) rd

  9. Indefinite Integral The indefinite integral of f(x), denoted by where C is an arbitrary constant is the most general integral of f(x) The indefinite integral of f(x) = 3x2 is rd

  10. A Strategy Guess and Test the integrand …or use a table of integrals rd

  11. Methods Of Integration Integrating Power Functions Fundamental Arithmetic Integration Rules Basic Integration Formulas Tables of Integrals Non-Integrability Partial Fractions Integration by Parts xe-xdx = -xe-x + e-x dx let u = x; dv = e-xdx du = 1; v = -e-x rd

  12. Integration by Parts d(uv) = udv + vdu udv = uv - vdu Show that xnexdx = xnex - nxn-1exdx + C let u = xn; dv = exdx then du = nxn-1dx; v = ex + C Thus, xnexdx = xnex - nxn-1exdx + C rd

  13. The top five rd

  14. Basic Rules of Integration rd

  15. The top four and the basic rules in action… rd

  16. Initial Conditions The rate at which annual income (y) changes with respect to years of education (x) is given by where y = 28,720 when x = 9. Find y. rd

  17. Integrating au , a > 0 rd

  18. Helpful Methods rd

  19. Use some algebra rd

  20. Adjusting for “du” – method of substitution rd

  21. More du’s rd

  22. Integrate x(x - 1)1/2 rd

  23. Partial Fractions rd

  24. Integration by Parts derived from the product rule for derivatives rd

  25. Another one? rd

  26. Integration by Tables A favorite integration formula of engineering students is: rd

  27. Check it out! rd

  28. Another Table Problem rd

  29. An Engineer’s Favorite Table rd

  30. The Definite Integral Areas under the curve rd

  31. Definite Integral Given a function f(x) that is continuous on the interval [a,b] we divide the interval into n subintervals of equal width, x, and from each interval choose a point, xi*.  Then the definite integral of f(x) from a to b is rd

  32. Area under the curve f(x) x x rd

  33. The Fundamental Theorem of Calculus • Let f be a continuous real-valued function defined • on a closed interval [a, b]. Let F be a function such that •     for all x in [a, b] • then •                                  . rd

  34. Fundamental Theorem rd

  35. Evaluating a definite integral rd

  36. Changing Limits rd

  37. The Area under a curve The area under the curve of a probability density function over its entire domain is always equal to one. Verify that the following function is a probability density function: rd

  38. Area between Curves Find the area bounded by y = 4 – 4x2 and y = x2 - 1 y1 = 4 – 4x2 y1 - y2 = 5 – 5x2 (-1, 0) (1, 0) y2 = x2 - 1 rd

  39. Area Find area bounded by y2 – x = 0 and y – x + 6 = 0. Curves intersection at y2 – y – 6 = 0; (y-3)(y+2) (x – 6)2 - x = 0 => (x – 9)(x – 4) = 0 (9,3) (4, -2) rd

  40. Rectilinear Motion A particle moves right from the origin on the x-axis with acceleration a = 5 – 2t and v0 = 0. How far does it go? a = 5 – 2t => v = 5t – t2 + v0 => s = 5t2/2 – t3/3 v = 0 when = 5t – t2 = 0 or when t = 5 s(5) = 125/2 -125/3 = 125/6 ft rd

  41. Y = x; x2; x3 rd

  42. Improper Integrals rd

  43. Example – an Improper Integral rd

  44. Let’s do another one… rd

  45. The Engineers Little Table of Improper Definite Integrals rd

  46. Some Applications Taking it to the limit… rd

  47. The Crime Rate The total number of crimes is increasing at the rate of 8t + 10 where t = months from the start of the year. How many crimes will be committed during the last 6 months of the year? rd

  48. Learning CurvesCumulative Cost hours to produce ith unit cumulative direct labor hrs to produce x units average unit hours to produce x units rd

  49. Learning CurvesApproximate Cumulative Cost rd

  50. Learning Curves - example Production of the first 10 F-222’s, the Air Force’s new steam driven fighter, resulted in a 71 percent learning curve in dollar cost where the first aircraft cost $18 million. What will be cost of the second lot of 10 aircraft? (sim-lc 18e6 20 71) rd

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