# 11X1 T16 01 area under curve (2011)

Post on 04-Jul-2015

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- 1. IntegrationArea Under Curve

2. IntegrationArea Under Curve 3. IntegrationArea Under Curve 4. IntegrationArea Under Curve Area 11 12 33 5. IntegrationArea Under Curve Area 11 12 33 Area 9 6. IntegrationArea Under Curve Area 11 12 33 Area 9 7. IntegrationArea Under Curve 10 11 Area 11 12 3333Area 9 8. IntegrationArea Under Curve 10 11 Area 11 12 33331 Area 9 9. IntegrationArea Under Curve 10 11 Area 11 12 33331 Area 9 Estimate Area 5 unit 2 10. IntegrationArea Under Curve 10 11 Area 11 12 33331 Area 9 Estimate Area 5 unit 2 Exact Area 4 unit 2 11. Area 0.40.43 0.83 1.23 1.63 23 12. Area 0.40.43 0.83 1.23 1.63 23 Area 5.76 13. Area 0.40.43 0.83 1.23 1.63 23 Area 5.76 14. 0.403 0.43 0.83 1.23 1.63 Area 0.40.43 0.83 1.23 1.63 23 Area 5.76 15. 0.403 0.43 0.83 1.23 1.63 Area 0.40.43 0.83 1.23 1.63 23 2.56 Area 5.76 16. 0.403 0.43 0.83 1.23 1.63 Area 0.40.43 0.83 1.23 1.63 23 2.56 Area 5.76 Estimate Area 4.16 unit 2 17. Area 0.20.23 0.43 0.63 0.83 13 1.23 1.43 1.63 1.83 23 18. Area 0.20.23 0.43 0.63 0.83 13 1.23 1.43 1.63 1.83 23 Area 4.84 19. Area 0.20.23 0.43 0.63 0.83 13 1.23 1.43 1.63 1.83 23 Area 4.84 20. 0.203 0.23 0.43 0.63 0.83 13 1.23 1.43 1.63 1.83 Area 0.20.23 0.43 0.63 0.83 13 1.23 1.43 1.63 1.83 23 Area 4.84 21. 0.203 0.23 0.43 0.63 0.83 13 1.23 1.43 1.63 1.83 Area 0.20.23 0.43 0.63 0.83 13 1.23 1.43 1.63 1.83 23 3.24 Area 4.84 22. 0.203 0.23 0.43 0.63 0.83 13 1.23 1.43 1.63 1.83 Area 0.20.23 0.43 0.63 0.83 13 1.23 1.43 1.63 1.83 23 3.24 Area 4.84Estimate Area 4.04 unit 2 23. As the widths decrease, the estimate becomes more accurate, letsinvestigate one of these rectangles.yy = f(x)x 24. As the widths decrease, the estimate becomes more accurate, letsinvestigate one of these rectangles.yy = f(x)x 25. As the widths decrease, the estimate becomes more accurate, letsinvestigate one of these rectangles.yy = f(x)c x A(c) is the area from 0 to c 26. As the widths decrease, the estimate becomes more accurate, letsinvestigate one of these rectangles.yy = f(x)cxx A(c) is the area from 0 to c A(x) is the area from 0 to x 27. A(x) A(c) denotes the area from c to x, and can be estimated bythe rectangle; 28. A(x) A(c) denotes the area from c to x, and can be estimated by the rectangle;f(x)x-c 29. A(x) A(c) denotes the area from c to x, and can be estimated by the rectangle;f(x) x-cA x Ac x c f x 30. A(x) A(c) denotes the area from c to x, and can be estimated by the rectangle;f(x) x-cA x Ac x c f x A x Ac f x xc 31. A(x) A(c) denotes the area from c to x, and can be estimated by the rectangle;f(x) x-cA x Ac x c f x A x Ac f x xc Ac h Ac h = width of rectangle h 32. A(x) A(c) denotes the area from c to x, and can be estimated by the rectangle;f(x) x-cA x Ac x c f x A x Ac f x xc Ac h Ac h = width of rectangle hAs the width of the rectangle decreases, the estimate becomes moreaccurate. 33. i.e. as h 0, the Area becomes exact 34. i.e. as h 0, the Area becomes exact Ac h Ac f x limh 0h 35. i.e. as h 0, the Area becomes exactAc h Ac f x lim h 0 hA x h A x limh 0 as h 0, c x h 36. i.e. as h 0, the Area becomes exactAc h Ac f x lim h 0 hA x h A x limh 0 as h 0, c x h A x 37. i.e. as h 0, the Area becomes exactAc h Ac f x lim h 0 hA x h A x limh 0 as h 0, c x h A x the equation of the curve is the derivative of the Area function. 38. i.e. as h 0, the Area becomes exactAc h Ac f x lim h 0 hA x h A x limh 0 as h 0, c x h A x the equation of the curve is the derivative of the Area function.The area under the curve y f x between x a and x b is; 39. i.e. as h 0, the Area becomes exactAc h Ac f x lim h 0 hA x h A x limh 0 as h 0, c x h A x the equation of the curve is the derivative of the Area function.The area under the curve y f x between x a and x b is;b A f x dxa 40. i.e. as h 0, the Area becomes exactAc h Ac f x lim h 0 hA x h A x limh 0 as h 0, c x h A x the equation of the curve is the derivative of the Area function.The area under the curve y f x between x a and x b is;b A f x dxa F b F a 41. i.e. as h 0, the Area becomes exactAc h Ac f x lim h 0 hA x h A x limh 0 as h 0, c x h A x the equation of the curve is the derivative of the Area function.The area under the curve y f x between x a and x b is;b A f x dxa F b F a where F x is the primitive function of f x 42. e.g. (i) Find the area under the curve y x 3 , between x = 0 and x= 2 43. e.g. (i) Find the area under the curve y x 3 , between x = 0 and 2 x= 2 A x 3 dx0 44. e.g. (i) Find the area under the curve y x 3 , between x = 0 and 2 x= 2 A x 3 dx0 2 x4 14 0 45. e.g. (i) Find the area under the curve y x 3 , between x = 0 and 2 x= 2 A x 3 dx02 x4 14 0 2 04 1 44 46. e.g. (i) Find the area under the curve y x 3 , between x = 0 and 2 x= 2 A x 3 dx02 x4 14 0 2 04 1 44 4 units 2 47. e.g. (i) Find the area under the curve y x 3 , between x = 0 and 2 x= 2 A x 3 dx02 x4 14 0 2 04 1 44 4 units 23ii x 2 1dx2 48. e.g. (i) Find the area under the curve y x 3 , between x = 0 and 2 x= 2 A x 3 dx 0 2 x4 1 4 0 2 04 1 4 4 4 units 23 3ii x 1dx 2 1 x 3 x 2 3 2 49. e.g. (i) Find the area under the curve y x 3 , between x = 0 and 2 x= 2 A x 3 dx02 x4 14 0 2 04 1 44 4 units 233ii x 1dx 2 1 x 3 x 2 3 21 33 3 1 2 3 2 3 3 50. e.g. (i) Find the area under the curve y x 3 , between x = 0 and 2 x= 2 A x 3 dx02 x4 14 0 2 04 1 44 4 units 233ii x 1dx 2 1 x 3 x 2 3 21 33 3 1 2 3 2 3 3 22 3 51. 5iii x 3dx 4 52. 55 1 2 iii x dx x 3 4 2 4 53. 55 1 2 iii x dx x 3 4 2 411 1 2 22 5 4 9 800 54. 55 1 2 iii x dx x 3 4 2 4 2 21 1 1 2 5 4 9 800 Exercise 11A; 1Exercise 11B; 1 aefhi, 2ab (i,ii), 3ace, 4b, 5a, 7*