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Substitution Method

Substitution Method. Integration. When one function is not the derivative of the other e.g. x is not the derivative of (4 x -1) and x is a variable. Substitute. Example 2. x - 1 is not the derivative of x +4 and it contains a variable. Substitute.

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Substitution Method

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  1. Substitution Method Integration

  2. When one function is not the derivative of the other e.g. x is not the derivative of (4x -1) and x is a variable Substitute

  3. Example 2 x - 1 is not the derivative of x +4 and it contains a variable Substitute

  4. Integrating and substituting back in for u

  5. Delta Exercise 12.8

  6. The definite integral

  7. Example 1 As 2x is the derivative, use inverse chain rule to integrate Substitute x = 4 Substitute x = 2

  8. Example 2 4x divided by 2x = 2 Solving x = 1/2 Substitute x = 1/2 into 4x + 3 to get 5 Divide the top by the bottom

  9. Example 3 Use substitution Substituting

  10. Delta Exercise 12.9

  11. Areas under curves

  12. To find the area under the curve between a and b…

  13. …we could break the area up into rectangular sections. This would overestimate the area.

  14. …or we could break the area up like this which would underestimate the area.

  15. The more sections we divide the area up into, the more accurate our answer would be.

  16. If each of our sections was infinitely narrow, we would have the area of each section as y The total area would be the sum of all these areas between a and b.

  17. is the sum all the areas of infinitely narrow width, dx and height, y.

  18. As the value of dx decreases, the area of the rectangle approaches y x dx y 0 dx

  19. The area of this triangle is 3 units squared The equation of the line is 2 If we sum all rectangles y 0 dx 3

  20. The area of this triangle is 3 units squared The equation of the line is dx 3 0 If we sum all rectangles y The area is 3 but the integral is -3 2

  21. http://rowdy.mscd.edu/~talmanl/MathAnim.html

  22. 2011 Level 2

  23. 2011 Level 2

  24. 2010 Level 2

  25. 2010 Level 2 • Area cannot be negative • Area = 6.67 units2

  26. -1 -6 8 Combination Integral is positive Integral is negative To find the area under the curve, we must integrate between -6 and -1 and between 8 and -1 separately and add the positive values together.

  27. -1 8 -6

  28. 2011 Level 2

  29. 2011 Level 2

  30. 2010 Question 1c

  31. 2010 Question 1c

  32. 2012

  33. 2012

  34. 2012

  35. 2012 • First find the x-value of the intersection point

  36. 2012

  37. 2010 Question 1e

  38. 2010 Question 1e • Find intersection points

  39. 2010 Question 1e

  40. Looking at areas a different way

  41. As the value of dy decreases, the area of the rectangle approaches x x dy The equation of the line is 3 Rearrange dy x 0 4 Definite Integral is

  42. Areas between two curves

  43. A typical rectangle in the upper section Solving these Equations gives y = 1 1 x = y x - x dy Area =(x - x )dy Area for this section is

  44. A typical rectangle in the lower section x = y x - x dy Area =(x - x )dy Area for this section is Total area is equal to 1

  45. Example 2 A typical rectangle dx y - y Area = (y - y)dx 0.707 Area

  46. Practice

  47. More practice

  48. Delta Exercise 16.2, 16.3, 16.4Worksheet 3 and 4

  49. Area in polar: extra for experts

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