1. 6.3 Volume by Slices Thurs April 9 Do Now Evaluate each integral 1) 2)

2. HW Review: worksheet480 #3-8 21 22 3) 621a) 263/4 = 65.75 4) 1.2521b) 31 5) 2/pi =.63722a) 3.5 6).95522b) -.373 7).582 8) 1.775

3. The Volume Problem What objects do we know the volume of? What kind of objects don’t we know the volume of?

4. Volumes by Slicing Let’s place a volume on the x-axis on its side. Then the x value becomes the height By slicing the volume into prisms, we get

5. Solids of Revolution Suppose f(x) is greater than 0 and continuous on the interval [a, b]. We can create a solid of revolution by revolving f(x) around the x-axis. Common solids of revolutions are cones, and cylinders

6. Volumes by Disks Suppose f(x) > 0 and continuous on the interval [a,b], and a solid of revolution is created by revolving around the x-axis. Then the volume of the solid is

7. Volumes by Disks 1) Draw the region 2) Determine the radius (f(x)) 3) Determine the bounds (lowest and highest x values) 4) Integrate the volume using disks

8. Ex 2.3 Suppose a region is bounded by x = 0, x = 12, y = 0, and y = 1/3x + 1, and is revolved about the x-axis. Find the volume of this solid

9. Ex 2.4 Revolve the region under the curve on the interval [0,4] about the x-axis and find the volume of this solid.

10. Closure Hand in: Find the volume of the region bounded by y = x + 1, y = 0, and x = 3, and revolved about the x-axis HW: p.381 #1-11 odds, 33

11. Method of Disks Using the Y Variable Fri April 10 Do Now Revolve the region under the curve bounded by y = 0, x = 0, and x = 4 about the x-axis, and find the volume of this solid.

12. HW Review: p.381 #1-11 33 1) 21pi = 65.973 3) 21/2 pi = 10.5pi = 32.987 5) 81/10 pi = 8.1pi = 25.447 7) 24573/13 pi = 1890.23pi = 5938.332 9) pi = 3.142 11) 3.19pi = 10.022 33) 376/15 pi = 25.067pi = 78.75

13. Using Y as the Independent Variable Suppose g(y) > 0 and g is continuous on the interval [c,d]. Then, revolving the region bounded by the curve x = g(y) about the y-axis generates a solid whose volume is

14. X or Y as our variable of integration? The decision on which variable to integrate with depends on the axis with which we are revolving the solid around. Revolve around x-axis (horizontal line) –Integrate with respect to x (y = f(x)) Revolve around y-axis (vertical line) –Integrate with respect to y (x = g(y))

15. Revolving About the Y-axis 1) Draw the region 2) Rewrite the function in terms of y 3) Determine the Y-bounds 4) Integrate the volume using disks

16. Ex 2.5 Find the volume of the solid resulting from revolving the portion of the curve from x = 0 to x = 2 about the y-axis

17. Ex 2.6a Let R be the region bounded by the graphs of, x = 0, and y = 1. Compute the volume of the solid formed by revolving R about the y-axis

18. Closure Journal Entry: Describe how we would find the volume of a solid revolved about the x-axis. What is the difference if we revolve about the y-axis? HW: p.381 #21 25 30 42 47

19. 6.3 Volumes by Washers Mon April 13 Do Now Find the volume of each region 1) Bounded by, y = 0 and x = 2 revolved about the x-axis 2) Bounded by, x = 0 and y = 4 revolved about the y-axis

20. HW Review: p.381 #21 25 30 42 47 21) 15/2 pi = 7.5pi = 23.562 25) 32pi = 100.531 30) 8pi = 25.133 42) 250pi = 785.398 47) 32/35 pi =.914pi = 2.871

21. Volumes with space between the region and the axis When there is a space between the region and axis of revolution, we cannot use disks. We need to subtract the extra space.

22. Washer Method Suppose f(x) is continuous on the interval [a,b] and does not cross the axis of revolution, but creates a cavity inside the solid when revolved, then

23. Washer Method 1) Draw the region 2) Determine axis of revolution and rewrite functions if necessary (x or y) 3) Determine the two radii of the solid 4) Determine bounds (x or y) 5) Set up integral using washer method 6) Integrate

24. Ex 2.6b Let R be the region bounded by the graphs of, x = 0 and y = 1. Compute the volume of the solid formed by revolving R about the x-axis.

25. Ex Let R be the region bounded by the graphs y = x^2, x = 0, and y = 8. Find the volume of the solid formed by revolving R about the x-axis

26. Closure Hand in: Let R be the region bounded by the graphs, x = 0, and y = 2. Find the volume of the solid formed by revolving R about the x-axis HW: p.381 #15 17 23 27 36 41

27. 5.2 Washer + Disk Practice Tues April 14 Do Now Let R be the region bounded by y = 2x, y = 0, and x = 6. Find the volume of the solid formed by revolving R about the y- axis

28. HW Review: p.381 #15 17 23 27 36 41 15) 256pi = 804.248 17) 656/3 pi = 218.67pi = 686.972 23) 3/10 pi = 0.3pi =.942 27) 704/15 pi = 46.93pi = 147.435 36) 8pi + 8pi = 16pi = 50.265 41) 1400/3 pi = 466.67pi = 1466.087

29. Practice Worksheet(blue) p.456 #1-4

30. Closure Journal Entry: Describe how to find a volume using the washer method. What is the difference between the washer method and the disk method? HW: Finish worksheet p.456 #1-4

31. 5.2 Washer + Disk Practice Wed April 15 Do Now Find the volume of the solid generated when the region enclosed by y = x^2 y = 2, and x = 0 is revolved about the y- axis

32. HW Review: Worksheet p.456 #1-4 1) 8pi = 25.133 2) 38pi/15 = 7.959 3) 13pi/6 = 6.807 4) 9pi/2 = 14.137

33. Worksheet Practice Worksheet (blue) p.456-457 #7-13 odds, 19, 20, 23, 25

34. Closure Journal Entry: Explain the difference between rotating about the y-axis, and rotating about the x-axis. Does the solid look different? How? HW: Finish worksheet p.456-457 #7-13 odds, 19, 20, 23, 25

35. 6.3 Revolving a Region about Different Lines Thurs April 16 Do Now Find the volume of the solid bounded by the functions y = x, x = 2, and y = 0, revolved about the y-axis

36. HW Review: Worksheet p.456 #7-13 odds 19, 20, 23, 25 7) =.920 9) 256pi/3 = 268.083 11) 2048pi/15 = 428.932 13) 4pi = 12.566 19) 8pi = 25.133 20) 8pi = 25.133 23) 72pi/5 = 45.239 25) = 10.036

37. Revolving about other lines When revolve about a different line, we want to consider if the line is horizontal or vertical –Horizontal: use x bounds –Vertical: use y bounds

38. Radii To calculate the radii, subtract the function from the center

39. Revolving a Region about Different Lines 1) Draw the picture 2) Identify whether the line is horizontal (use x) or vertical (use y) 3) Identify the outer radius –The curve furthest away from revolving line –Distance between radius and axis of revolution –(function - center) 4) Identify the inner radius –The curve closer to revolving line –Distance between radius and axis of revolution –(function - center) 5) Identify the bounds (x or y) 6) Use washers or disks to find the volume

40. Ex 2.6c Let R be the region bounded by the graphs of, x = 0 and y = 1. Compute the volume of the solid formed by revolving R about the line y = 2.

41. Closure Hand in:Let R be the region bounded by the graphs of and y = 0. Compute the volume of the solid formed by revolving R about the line y = 7. HW: p.381 #13 29 31 34 35 39 45

42. 6.3 Revolving about Different Lines Fri April 17 Do Now Find the volume of the solid generated by revolving the region bounded by about the x-axis

43. HW Review: p.381 #13 29 31 34 35 39 45 13) (iv) 29) 128/5 pi = 80.425 31) 40pi = 125.663 34) 776/15 pi = 162.525 35) 824/15 pi = 172.578 39) 1872/5 pi = 1176.212 45) 96/5 pi = 60.319

44. Revolving a Region about Different Lines 1) Draw the picture 2) Identify whether the line is horizontal (use x) or vertical (use y) 3) Identify the outer radius –The curve furthest away from revolving line –Distance between radius and axis of revolution –(top curve - bottom curve) 4) Identify the inner radius –The curve closer to revolving line –Distance between radius and axis of revolution –(top curve - bottom curve) 5) Identify the bounds (x or y) 6) Use washers or disks to find the volume

45. Things to Note The outer radius is the function furthest away from the axis of revolution (center) –Distance between outer radius and axis of revolution The inner radius is the function closer to the axis of revolution (center) –Distance between inner radius and axis of revolution –If the inner radius is the axis of revolution, then you use the disk method –If the inner radius is another function, then use the washer method

46. You try Let R be the region bounded by the graphs of and y = 0. Compute the volume of the solid formed by revolving R about the line y = -3.

47. Practice Worksheet(green) p.424 #21-24

48. Closure Journal Entry: Draw an example of a solid that would need the disk method. Draw an example of a solid that would need the washer method. What is the difference? HW: worksheet p.424 #21-24 6.2-6.3 Quiz Thurs April 23

49. 6.3 Revolving About Different Lines Mon April 20 Do Now Find the volume of the solid generated by revolving the region bounded by y = 2, y = 2sinx, x = 0, and x = pi/2, about the line y = 2

50. HW Review worksheet p.424 21a) 8pi/3 (8.38)b) 28pi/3 (29.32) 22a) 32pi/5 (20.12) b) 224pi/15 (46.91) 23a) 32pi/5 (20.12) b) 224pi/15 (46.91) 24a) 2pi/3 (2.09)b) 2pi/3 (2.09)

51. Reminder When revolving about a horizontal line, use x When revolving about a vertical line, use y When revolving about a line, subtract that value from each radius

52. Practice (green) worksheet P.424 #28ab, 29cdef

53. Closure Hand in: Let R be the region bounded by y = x, y = -x, and x = 1. Compute the volume of the solid formed by revolving R about the line y = 1 HW: P.424 #26ab 28ab, 29cdef 6.2 6.3 Quiz Thurs April 23

54. 6.2 6.3Review Tues April 21 Do Now Let R be the region bounded by y = x, y = -x, and x = 1. Compute the volume of the solid formed by revolving R about the line y = -1

55. HW Review: p.424 #26ab, 28ab 26a) = 3.82 29f) 36pi = 113.1 26b)= 8.75 28a) pi/10 =.314 28b) 9pi/14 = 2.02 29c) 18pi = 56.549 29d) 36pi = 113.097 29e) 18pi = 56.549

56. 6.2 6.3 Quiz Review Average Value –Divide integral by length of interval Volumes –Disk Method –Washer Method –Revolving about x-axis –Revolving about y-axis –Revolving about other lines

57. More Practice P.424 #30acd, 31cdef

58. HW Review: p.424 #30acd, 31cdef 30a) 512pi/15 = 107.233 30c) 384pi/5 = 241.274 30d) 1408pi/15 = 294.891 31c) pi/6 = 0.524 31d) 7pi/15 = 1.466 31e) 7pi/6 = 3.665 31f) 13pi/15 = 2.723

59. Closure Journal Entry: When finding volumes, how do we know whether to integrate with respect to x or y? Do we have a choice? 6.2 6.3 Quiz Thurs

60. 6.2 6.3 Quiz Review Wed April 22 Do Now Let R be the region bounded by y = x, y = -x, and x = 1. Compute the volume of the solid formed by revolving R about the line y = 1

61. HW Review: Worksheet (oj)p.407 #7-18 7) 2pi/315) 2pi/3 8) 6pi16) pi 9) 4 - pi17) 117pi/5 11) 32pi/518) 108pi/5 12) 128pi/7 13) 36pi 14) pi/30