1. Integral Calculus
Problems
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2. 2-Variable Function with a Maximum
z = f(x,y)
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3. Sequence Problem Solving
?? ? ?
77 7*7 = 49 Skip every other prime 31 ?

4. 2-Variable Function with both Maxima and Minima
z = f(x,y)
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5. 2-Variable Function with a Saddle Point
z = f(x,y)
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6. Natural Logarithms With x/h = n tending to infinity yielding e
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7. Integration Integral Calculus Stochastic (Probability) Models
Differential Equations
Dynamic Models
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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)
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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
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10. …or use a table of integrals
A Strategy
Guess and Test the integrand
…or use a table of integrals
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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-x dx du = 1; v = -e-x
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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-1ex dx + C
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13. The top five
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14. Basic Rules of Integration
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15. The top four and the basic rules in action…
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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.
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17. Integrating au , a > 0
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18. Helpful Methods
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19. Use some algebra
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20. Adjusting for “du” – method of substitution
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21. More du’s
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22. Integrate x(x - 1)1/2
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23. Partial Fractions
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24. Integration by Parts derived from the product rule for derivatives
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25. Another one?
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26. Integration by Tables
A favorite integration formula of engineering students is:
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27. Check it out!
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28. Another Table Problem
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29. An Engineer’s Favorite Table
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30. The Definite Integral
Areas under the curve
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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
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32. Area under the curve
f(x) x x
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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
                                 .
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34. Fundamental Theorem
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35. Evaluating a definite integral
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36. Changing Limits
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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:
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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
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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)
(+ 32/3 54/3 -81/ / ) -->125/6
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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
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41. Y = x; x2; x3
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42. Improper Integrals
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43. Example – an Improper Integral
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44. Let’s do another one…
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45. The Engineers Little Table of Improper Definite Integrals
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46. Some Applications
Taking it to the limit…
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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?
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48. Learning Curves Cumulative Cost
hours to produce ith unit
cumulative direct labor
hrs to produce x units
average unit hours to
produce x units
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49. Learning Curves Approximate Cumulative Cost
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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 18e )
(sim-lc 18e )
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51. The Average of a Function
The average or mean value of a function y = f(x) over the interval [a,b] is given by:
Find the average of the function y = x2 over the interval [1,3]:
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52. Average profit
An oil company’s profit in dollars for the qth million gallons sold is given by
P = P(q) = 369q – 2.1q2 – 400
If the company sells 100 million gallons this year, what is the average profit per gallon sold?
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53. An Inventory Problem
Demand for an item is constant over time at the rate of 720 per year. Whenever the on-hand inventory reaches zero, a shipment of 60 units is received. The inventory holding cost is based upon the average on-hand inventory.
Let y = 60 – 720t be the on-hand inventory as a function of time where t is in years. It takes 60/720 = 1/12 yr to go from an inventory of 60 to 0. 60 t
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54. Annuities A is the present value of a continuous income stream
Let A = the present value of a continuous annuity at an annual rate r (compounded continuously) for T years if a payment at time t is at the rate of f(t) per year. Then
A is the present value of a continuous income stream
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55. Annuity Example
Determine the present value of a continuous annuity at an annual rate of 8% for 10 years if the payment at time t is at the rate of 1000t dollars per year.
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56. Annuities
Let S = the accumulated amount of a continuous annuity at an annual rate r (compounded continuously) for T years if a payment at time t is at the rate of f(t) per year. Then
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57. Back to the example
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58. More of that example
Recall continuous compounding
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59. Iterated Integrals
Evaluate
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60. Double Integral
Evaluate the integral over R where R is the triangle formed by y = x, y = 0, x = 1.
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61. f(x, y) = 2 for 0 < x < y < 1
Find
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62. Area between Regions
Find the area bounded by y = x3 - 4x and y = 3x Curves intersect at (cubic )  ( )
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63. Quadratic Equation Derivation
ax2 + bx + c = 0 x2 + (b/a)x + (c/a) = 0 Complete the square x2 + (b/a)x + (c/a) + (b2/4a2) = (b2/4a2) (x + b/2a)2 = -c/a + b2/4a2 x = -b/2a + (- c/a + b2/4a2)1/2 = - b  (b2 – 4ac)1/2 2a
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64. Partial Fractions
Solve for A and B by the cover-up method and integrate to get 3 ln(x - 2) + 2 ln(x + 4) + C or ln(x - 2)3(x + 4)2 + C
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65. Sketch the Plane
2x + 3y + 4 = 12 z 3 4 y 6 x
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66. Tangent
Find the equation of the tangent to the curve y = x ln x – x where x = 1. y' = 1 + ln x – 1 = ln x or ln 1 = 0 y = -1 at x = 1 with slope 0. (y +1) = 0(x – 1) => y = -1
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67. Predator/Prey
# prey attack y = K(1 - ex), K and  constants dy/dx = Kex and ex = -y/K +1 => dy/dx = K(-y/K + 1) = (K – y)
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68. Average Value of a Function over [a, b].
Find the average value of f(x) = 2 – 3x2 over [-1, 2]
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