2. www.mathsrevision.com Higher Higher Unit 2 www.mathsrevision.com What is Integration The Process of Integration Integration & Area of a Curve Exam Type Questions Higher Outcome 2

3. www.mathsrevision.com Higher Integration Part 1 Anti-Differentiation Integration can be thought of as the opposite of differentiation (just as subtraction is the opposite of addition). In general:DifferentiatingIntegrating Confusing? Is there any easier way? Outcome 2

4. www.mathsrevision.com Higher Differentiation multiply by power decrease power by 1 Integration increase power by 1 divide by new power Where does this + C come from? Integration Outcome 2

5. www.mathsrevision.com Higher Integrating is the opposite of differentiating, so: integrate But: differentiate integrate Integrating 6x ….......which function do we get back to? Integration Outcome 2

6. www.mathsrevision.com Higher Solution: When you integrate a function remember to add the Constant of Integration …………… + C Integration Outcome 2

7. www.mathsrevision.com Higher means “integrate 6x with respect to x” means “integrate f(x) with respect to x” Notation This notation was “invented” by Gottfried Wilhelm von Leibniz  Integration Outcome 2

8. www.mathsrevision.com Higher Examples: Integration Outcome 2

9. www.mathsrevision.com Higher Integration Outcome 2 Just like differentiation, we must arrange the function as a series of powers of x before we integrate.

10. www.mathsrevision.com Higher To get the function F(x) from the derivative F’(x) we do the opposite, i.e. we integrate. Hence : Integration Outcome 2

11. www.mathsrevision.com Higher Further examples of integration Exam Standard Integration Outcome 2

12. www.mathsrevision.com Higher The integral of a function can be used to determine the area between the x-axis and the graph of the function. NB:this is a definite integral. It has lower limit a and an upper limit b. Area under a Curve Outcome 2

13. www.mathsrevision.com Higher Examples: Area under a Curve Outcome 2

14. www.mathsrevision.com Higher Conventionally, the lower limit of a definite integral is always less then its upper limit. Area under a Curve Outcome 2

15. www.mathsrevision.com Higher a b cd y=f(x) Very Important Note: When calculating integrals: areas above the x-axis are positive areas below the x-axis are negative When calculating the area between a curve and the x-axis: make a sketch calculate areas above and below the x-axis separately ignore the negative signs and add Area under a Curve Outcome 2

16. www.mathsrevision.com Higher The Area Between Two Curves To find the area between two curves we evaluate: Area under a Curve Outcome 2

17. www.mathsrevision.com Higher Example: Area under a Curve Outcome 2

18. www.mathsrevision.com Higher Complicated Example: The cargo space of a small bulk carrier is 60m long. The shaded part of the diagram represents the uniform cross-section of this space. Find the area of this cross- section and hence find the volume of cargo that this ship can carry. Area under a Curve Outcome 2 9 1

19. www.mathsrevision.com Higher The shape is symmetrical about the y-axis. So we calculate the area of one of the light shaded rectangles and one of the dark shaded wings. The area is then double their sum. The rectangle: let its width be s The wing: extends from x = s to x = t The area of a wing (W ) is given by: Area under a Curve

20. www.mathsrevision.com Higher The area of a rectangle is given by: The area of the complete shaded area is given by: The cargo volume is: Area under a Curve Outcome 2

21. www.mathsrevision.com Higher Exam Type Questions Outcome 2 At this stage in the course we can only do Polynomial integration questions. In Unit 3 we will tackle trigonometry integration

22. Integration Higher Mathematics www.maths4scotland.co.uk Next

23. Calculus Revision Back Next Quit Integrate Integrate term by term simplif y

24. Calculus Revision Back Next Quit Find

25. Calculus Revision Back Next Quit Integrate Multiply out brackets Integrate term by term simplify

26. Calculus Revision Back Next Quit Find

27. Calculus Revision Back Next Quit Integrate Standard Integral (from Chain Rule)

28. Calculus Revision Back Next Quit Find p, given

29. Calculus Revision Back Next Quit Evaluate Straight line form

30. Calculus Revision Back Next Quit Find Use standard Integral (from chain rule)

31. Calculus Revision Back Next Quit Find Integrate term by term

32. Calculus Revision Back Next Quit Integrate Straight line form

33. Calculus Revision Back Next Quit Integrate Straight line form

34. Calculus Revision Back Next Quit Integrate Straight line form

35. Calculus Revision Back Next Quit Integrate Split into separate fractions

36. Calculus Revision Back Next Quit Find Use standard Integral (from chain rule)

37. Calculus Revision Back Next Quit Find

38. Calculus Revision Back Next Quit Find

39. Calculus Revision Back Next Quit Integrate Straight line form

40. Calculus Revision Back Next Quit Given the acceleration a is: If it starts at rest, find an expression for the velocity v where Starts at rest, so v = 0, when t = 0

41. Calculus Revision Back Next Quit A curve for which passes through the point Find y in terms of x. Use the point

42. Calculus Revision Back Next Quit Integrate Split into separate fractions Multiply out brackets

43. Calculus Revision Back Next Quit If passes through the point express y in terms of x. Use the point

44. Calculus Revision Back Next Quit Integrate Straight line form

45. Calculus Revision Back Next Quit The graph of passes through the point (1, 2). express y in terms of x. If simplify Use the point Evaluate c

46. Calculus Revision Back Next Quit Integrate Straight line form

47. Calculus Revision Back Next Quit A curve for which passes through the point (–1, 2). Express y in terms of x. Use the point

48. Calculus Revision Back Next Quit Evaluate Cannot use standard integral So multiply out

49. Calculus Revision Back Next Quit Evaluate Straight line form

50. Calculus Revision Back Next Quit Evaluate Use standard Integral (from chain rule)

51. Calculus Revision Back Next Quit The curve passes through the point Find f(x ) use the given point

52. Calculus Revision Back Next Quit Integrate Integrate term by term

53. Calculus Revision Back Next Quit Integrate Integrate term by term

54. Calculus Revision Back Next Quit Evaluate

55. Quit C P D www.maths4scotland.co.uk © CPD 2004