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

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

5. Integrating and substituting back in for u

6. Delta Exercise 12.8

7. The definite integral

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

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

10. Example 3 Use substitution Substituting

11. Delta Exercise 12.9

12. Areas under curves

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

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

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

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

17. 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.

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

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

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

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

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

23. 2011 Level 2

25. 2010 Level 2

26. Area cannot be negative Area = 6.67 units 2

27. 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. -6 8

28. -6 8

29. 2011 Level 2

31. 2010 Question 1c

33. 2012

36. First find the x-value of the intersection point

37. 2012

38. 2010 Question 1e

39. Find intersection points

40. 2010 Question 1e

41. Looking at areas a different way

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

43. Areas between two curves

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

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

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

47. Practice

48. More practice

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

50. Area in polar: extra for experts