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Integration

Integration. This is not your father’s area?. The economy is so bad that the following is happening with Snap, Crackle and Pop. They are thinking of replacing all three of them with Pow Kelloggs hired a “cereal” killer to kill them all Snap is spreading rumors that “Pop was a rolling stone”

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Integration

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  1. Integration This is not your father’s area? http://nm.mathforcollege.com

  2. The economy is so bad that the following is happening with Snap, Crackle and Pop • They are thinking of replacing all three of them with Pow • Kelloggs hired a “cereal” killer to kill them all • Snap is spreading rumors that “Pop was a rolling stone” • They are selling smack, crack and pot, respectively. http://nm.mathforcollege.com

  3. Ask me what I should already knowThe pre-requisite questions http://nm.mathforcollege.com

  4. The velocity of a body is given as v(t)=t2. Given that • 27 • 54 • 63 • Cannot be determined the location of the body at t=6 is http://nm.mathforcollege.com

  5. The velocity of a body is given as v(t)=t2. Given that • 27 • 54 • 63 • Cannot be determined the distance covered by the body between t=3 and t=6 is http://nm.mathforcollege.com

  6. The exact mean value of the function f(x) from a to b is http://nm.mathforcollege.com

  7. y 5 a b 2 d x 7 Given the f(x) vs x curve, and the magnitude of the areas as shown, the value of • -2 • 2 • 12 • Cannot be determined c http://nm.mathforcollege.com

  8. y 5 a b 2 d x 7 Given the f(x) vs x curve, and the magnitude of the areas as shown, the value of • -2 • 2 • 12 • Cannot be determined c http://nm.mathforcollege.com

  9. y 5 a b 2 c x 7 Given the f(x) vs x curve, and the magnitude of the areas as shown, the value of • 5 • 12 • 14 • Cannot be determined http://nm.mathforcollege.com

  10. y 5 a b 2 c x 7 Given the f(x) vs x curve, and the magnitude of the areas as shown, the value of • -7 • -2 • 12 • Cannot be determined http://nm.mathforcollege.com

  11. y 5 a b 2 c x 7 Given the f(x) vs x curve, and the magnitude of the areas as shown, the value of • -7 • -2 • 7 • 12 http://nm.mathforcollege.com

  12. The area of a circle of radius a can be found by the following integral http://nm.mathforcollege.com

  13. Physically, integrating means finding the • Area under the curve from a to b • Area to the left of point a • Area to the right of point b • Area above the curve from a to b http://nm.mathforcollege.com

  14. Physical examples http://nm.mathforcollege.com

  15. Distance Covered By Rocket http://nm.mathforcollege.com

  16. Concentration of Benzene u= velocity of ground water flow in the x -direction (m/s) D = dispersion coefficient (m2) C0= initial concentration (kg/m3) http://nm.mathforcollege.com

  17. Is Wal*** “short shifting” you? http://nm.mathforcollege.com

  18. Calculating diameter contraction for trunnion-hub problem http://nm.mathforcollege.com

  19. END http://nm.mathforcollege.com

  20. 7.02Trapezoidal Rule http://nm.mathforcollege.com

  21. The morning after learning trapezoidal rule… http://nm.mathforcollege.com

  22. You are happy because • You are thinking about your free will • You are delusional • You have to see my pretty face for only three more weeks • All of the above http://nm.mathforcollege.com

  23. Two-segment trapezoidal rule of integration is exact for integration of polynomials of order of at most • 1 • 2 • 3 • 4 http://nm.mathforcollege.com

  24. In trapezoidal rule, the number of segments needed to get the exact value for a general definite integral • 1 • 2 • 1 googol • infinite http://nm.mathforcollege.com

  25. In trapezoidal rule, the number of points at which function is evaluated for 8 segments is • 8 • 9 • 16 • 17 http://nm.mathforcollege.com

  26. In trapezoidal rule, the number of function evaluations for 8 segments is • 8 • 9 • 16 • 17 http://nm.mathforcollege.com

  27. The distance covered by a rocket from t=8 to t=34 seconds is calculated using multiple segment trapezoidal rule by integrating a velocity function. Below is given the estimated distance for different number of segments, n. • 1 • 2 • 3 • 4 The number of significant digits at least correct in the answer for n=5 is http://nm.mathforcollege.com

  28. The value of by using one segment trapezoidal rule is most nearly • 201.33 • 212.00 • 424.00 • 742.00 http://nm.mathforcollege.com

  29. The velocity vs time is given below. A good estimate of the distance in meters covered by the body between t=0.5 and 1.2 seconds is • 213*0.7 • 256*0.7 • 256*1.2-213*0.5 • ½*(213+256)*0.7 http://nm.mathforcollege.com

  30. 7.05Gauss Quadrature Rule http://nm.mathforcollege.com

  31. The morning after learning Gauss Quadrature Rule… http://nm.mathforcollege.com

  32. Autar Kaw is looking for a stage name. Please vote your choice. • The last mindbender • Қ (formerly known as Kaw) • Kid Cuddi • Kaw & Saki http://nm.mathforcollege.com

  33. is exactly http://nm.mathforcollege.com

  34. A scientist would derive one-point Gauss Quadrature Rule based on getting exact results of integration for function f(x)=a0+a1x. The one-point rule approximation for the integral is http://nm.mathforcollege.com

  35. For integrating any first order polynomial, the one-point Gauss quadrature rule will give the same results as • 1-segment trapezoidal rule • 2-segment trapezoidal rule • 3-segment trapezoidal rule • All of the above http://nm.mathforcollege.com

  36. A scientist can derive a one-point quadrature rule for integrating definite integrals based on getting exact results of integration for the following function • a0+a1x+a2x2 • a1x+a2x2 • a1x • a2x2 http://nm.mathforcollege.com

  37. For integrating any third order polynomial, the two-point Gauss quadrature rule will give the same results as • 1-segment trapezoidal rule • 2-segment trapezoidal rule • 3-segment trapezoidal rule • None of the above http://nm.mathforcollege.com

  38. The highest order of polynomial for which the n-point Gauss-quadrature rule would give an exact integral is • n • n+1 • 2n-1 • 2n http://nm.mathforcollege.com

  39. You are asked to estimate the water flow rate in a pipe of radius 2m at a remote area location with a harsh environment. You already know that velocity varies along the radial location, but do not know how it varies. The flow rate, Q is given by • 0, 2 • 1, 2 • 0, 1 • 0.42, 1.58 To save money, you are allowed to put only two velocity probes (these probes send the data to the central office in New York, NY via satellite) in the pipe. Radial location, r is measured from the center of the pipe, that is r=0 is the center of the pipe and r=2m is the pipe radius. The radial locations you would suggest for the two velocity probes for the most accurate calculation of the flow rate are http://nm.mathforcollege.com

  40. A scientist an approximate formula for integration as ,where The values of c1 and x1 are found by assuming that the formula is exact for the functions of the form a0x+a1x2 polynomial. Then the resulting formula would be exact for integration. http://nm.mathforcollege.com

  41. END http://nm.mathforcollege.com

  42. most nearly is The exact value of • 7.8036 • 11.807 • 14.034 • 19.611 http://nm.mathforcollege.com

  43. Velocity distribution of a fluid flow through a pipe varies along the radius, and is given by v(r). The flow rate through the pipe of radius a is given by http://nm.mathforcollege.com

  44. The value of the integral • x3 • x3 +C • x3/3 • x3/3 +C • 2x http://nm.mathforcollege.com

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