1. Volume of Disks & Washers
Section 7.2
Calculus AP/Dual, Revised ©2017
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§7.2: Volume of Disks and Washers

2. §7.2: Volume of Disks and Washers
Review
The oil drum has a height of 5 feet and a radius of 2 feet. Find the volume.
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§7.2: Volume of Disks and Washers

3. §7.2: Volume of Disks and Washers
Calculus in Motion
Link – Calc in Motion
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§7.2: Volume of Disks and Washers

4. §7.2: Volume of Disks and Washers
Animation
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§7.2: Volume of Disks and Washers

5. §7.2: Volume of Disks and Washers
Cutting Up a Lime
RADIUS
HEIGHT
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§7.2: Volume of Disks and Washers

6. Graphs Rotate Around 𝒙–Axis or Any Horizontal Line
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§7.2: Volume of Disks and Washers

7. Graphs Rotate Around 𝒚–Axis or Any Vertical Line
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§7.2: Volume of Disks and Washers

8. §7.2: Volume of Disks and Washers
YouTube Clip
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§7.2: Volume of Disks and Washers

9. §7.2: Volume of Disks and Washers
Calculus in Motion
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§7.2: Volume of Disks and Washers

10. §7.2: Volume of Disks and Washers
Steps for Disk Method
Draw the function
Identify the type of shape and what the graph is in relations to (respects to 𝒙 or 𝒚-axis)
Identify and/or solve for the intersection points
Solve for 𝑹.
Vertical Strip Equation: 𝑽=𝝅 𝒂 𝒃 𝑹 𝒙 𝟐 𝒅𝒙 where it rotates around the 𝒙–axis
Horizontal Strip Equation: 𝑽=𝝅 𝒄 𝒅 𝑹 𝒚 𝟐 𝒅𝒚 where it rotates around the 𝒚–axis
Simplify, Solve, and LABEL accordingly. DO NOT SEPARATE THE EQUATIONS
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§7.2: Volume of Disks and Washers

11. §7.2: Volume of Disks and Washers
Example 1
Find the volume of a solid formed by revolving 𝒚= 𝒙 , the 𝒙-axis and the line of 𝒙=𝟓 about the 𝒙-axis.
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§7.2: Volume of Disks and Washers

12. §7.2: Volume of Disks and Washers
Example 1
Find the volume of a solid formed by revolving 𝒚= 𝒙 , the 𝒙-axis and the line of 𝒙=𝟓 about the 𝒙-axis.
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§7.2: Volume of Disks and Washers

13. §7.2: Volume of Disks and Washers
Example 2
Find the volume of the solid of revolution generated by revolving the region bounded by the graph of 𝒚= 𝒙 𝟑 , the 𝒚-axis, and the lines 𝒚=𝟏 and 𝒚=𝟖 about the 𝒚-axis.
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§7.2: Volume of Disks and Washers

14. §7.2: Volume of Disks and Washers
Example 2
Find the volume of the solid of revolution generated by revolving the region bounded by the graph of 𝒚= 𝒙 𝟑 , the 𝒚-axis, and the lines 𝒚=𝟏 and 𝒚=𝟖 about the 𝒚-axis.
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§7.2: Volume of Disks and Washers

15. §7.2: Volume of Disks and Washers
Your Turn
Find the volume of the solid of revolution generated by revolving the region bounded by the graph of 𝒚= 𝟗− 𝒙 𝟐 in the first quadrant about the 𝒙-axis.
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§7.2: Volume of Disks and Washers

16. §7.2: Volume of Disks and Washers
Your Turn
Find the volume of the solid of revolution generated by revolving the region bounded by the graph of 𝒚= 𝟗− 𝒙 𝟐 in the first quadrant about the 𝒙-axis.
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§7.2: Volume of Disks and Washers

17. §7.2: Volume of Disks and Washers
Let’s Have Some Fun
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§7.2: Volume of Disks and Washers

18. §7.2: Volume of Disks and Washers
Example 3
Rotate the region enclosed by 𝒚= (𝐬𝐢𝐧 𝒙) 𝟏/𝟐 on the interval 𝟎≤𝒙≤𝝅 about the 𝒙–axis. Determine the volume of the solid formed.
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§7.2: Volume of Disks and Washers

19. §7.2: Volume of Disks and Washers
Example 4
Rotate the region enclosed by 𝒚= (𝐬𝐢𝐧 𝒙) 𝟏/𝟐 on the interval 𝟎≤𝒙≤𝝅 about the 𝒙–axis. Determine the volume of the solid formed. When this shape is rotated, what does it look like?
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§7.2: Volume of Disks and Washers

20. Slice the Item Rotating Around to 𝒙–Axis
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§7.2: Volume of Disks and Washers

21. §7.2: Volume of Disks and Washers
Example 4
Rotate the region enclosed by 𝒚= 𝒙 𝟐 on the interval 𝒙=𝟎 and 𝒙=𝟐 about the 𝒙–axis. Determine the volume of the solid formed.
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§7.2: Volume of Disks and Washers

22. §7.2: Volume of Disks and Washers
Example 4
Rotate the region enclosed by 𝒚= 𝒙 𝟐 on the interval 𝒙=𝟎 and 𝒙=𝟐 about the 𝒙–axis. Determine the volume of the solid formed.
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§7.2: Volume of Disks and Washers

23. §7.2: Volume of Disks and Washers
Example 4
Rotate the region enclosed by 𝒚= 𝒙 𝟐 on the interval 𝒙=𝟎 and 𝒙=𝟐 about the 𝒙–axis. Determine the volume of the solid formed.
What does it look like?
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§7.2: Volume of Disks and Washers

24. §7.2: Volume of Disks and Washers
Example 4
Rotate the region enclosed by 𝒚= 𝒙 𝟐 on the interval 𝒙=𝟎 and 𝒙=𝟐 about the 𝒙–axis. Determine the volume of the solid formed.
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§7.2: Volume of Disks and Washers

25. §7.2: Volume of Disks and Washers
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§7.2: Volume of Disks and Washers

26. §7.2: Volume of Disks and Washers
Determine 𝒅𝒙
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§7.2: Volume of Disks and Washers

27. §7.2: Volume of Disks and Washers
Your Turn
Find the volume of a solid of revolution generated by revolving the region bounded by the graph of 𝒚= 𝒙 𝟑 , the 𝒙–axis, and the lines of 𝒙=−𝟏 and 𝒙=𝟐 about the 𝒙–axis
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§7.2: Volume of Disks and Washers

28. §7.2: Volume of Disks and Washers
Example 5
Rotate the region enclosed by 𝒚= 𝟏 𝟐 𝒙−𝟏 on the interval 𝒚=𝟎, 𝒚=𝟐 and 𝒙=𝟎 about the 𝒚–axis. Determine the volume of the solid formed.
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§7.2: Volume of Disks and Washers

29. §7.2: Volume of Disks and Washers
Example 5
Rotate the region enclosed by 𝒚= 𝟏 𝟐 𝒙−𝟏 on the interval 𝒚=𝟎, 𝒚=𝟐 and 𝒙=𝟎 about the 𝒚–axis. Determine the volume of the solid formed.
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§7.2: Volume of Disks and Washers

30. §7.2: Volume of Disks and Washers
Example 5
Rotate the region enclosed by 𝒚= 𝟏 𝟐 𝒙−𝟏 on the interval 𝒚=𝟎, 𝒚=𝟐 and 𝒙=𝟎 about the 𝒚–axis. Determine the volume of the solid formed.
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§7.2: Volume of Disks and Washers

31. §7.2: Volume of Disks and Washers
Example 5
Rotate the region enclosed by 𝒚= 𝟏 𝟐 𝒙−𝟏 on the interval 𝒚=𝟎, 𝒚=𝟐 and 𝒙=𝟎 about the 𝒚–axis. Determine the volume of the solid formed.
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§7.2: Volume of Disks and Washers

32. Slice it in Revolving the 𝒚–Axis
Rotate the region enclosed by 𝒚= 𝟏 𝟐 𝒙−𝟏 on the interval 𝒚=𝟎, 𝒚=𝟐 and 𝒙=𝟎 about the 𝒚–axis. Determine the volume of the solid formed.
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§7.2: Volume of Disks and Washers

33. §7.2: Volume of Disks and Washers
Example 6
Rotate the region enclosed by 𝒚=𝟒− 𝒙 𝟐 about the 𝒚–axis in the first quadrant. Determine the volume of the solid formed.
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§7.2: Volume of Disks and Washers

34. §7.2: Volume of Disks and Washers
Example 6
Rotate the region enclosed by 𝒚=𝟒− 𝒙 𝟐 about the 𝒚–axis in the first quadrant. Determine the volume of the solid formed.
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§7.2: Volume of Disks and Washers

35. Slice it in Rotating the 𝒚–Axis
Rotate the region enclosed by 𝒚=𝟒− 𝒙 𝟐 about the 𝒚–axis in the first quadrant. Determine the volume of the solid formed.
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§7.2: Volume of Disks and Washers

36. §7.2: Volume of Disks and Washers
Your Turn
Rotate the region enclosed by 𝒚=𝟒− 𝒙 𝟐 about the 𝒙–axis in the first quadrant. Determine the volume of the solid formed.
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§7.2: Volume of Disks and Washers

37. §7.2: Volume of Disks and Washers
Your Turn
Rotate the region enclosed by 𝒚=𝟒− 𝒙 𝟐 about the 𝒙–axis in the first quadrant. Determine the volume of the solid formed.
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§7.2: Volume of Disks and Washers

38. Slice it in Rotating the 𝒙–Axis
Rotate the region enclosed by 𝒚=𝟒− 𝒙 𝟐 about the 𝒙–axis in the first quadrant. Determine the volume of the solid formed.
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§7.2: Volume of Disks and Washers

39. §7.2: Volume of Disks and Washers
The Washer Method
Draw the function
Identify the type of shape and what the graph is in relations to (respects to 𝒙 or 𝒚-axis)
Identify and/or solve for the intersection points
Solve for 𝑹 and 𝒓
Vertical Strip Equation: 𝑽=𝝅 𝒂 𝒃 𝑹(𝒙) 𝟐 − 𝒓(𝒙) 𝟐 𝒅𝒙 where it rotates around the 𝒙–axis
Horizontal Strip Equation: 𝑽=𝝅 𝒄 𝒅 𝑹(𝒚) 𝟐 − 𝒓(𝒚) 𝟐 𝒅𝒚 where it rotates around the 𝒚–axis
𝑹 is the top/right radius and r is the bottom/left of the radius
NOTE: Square the radius first. It is NOT 𝑹(𝒙) 𝟐 − 𝒓(𝒙) 𝟐
Simplify, Solve, and LABEL accordingly. DO NOT SEPARATE THE EQUATIONS
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§7.2: Volume of Disks and Washers

40. §7.2: Volume of Disks and Washers
Example 7
Find the volume 𝑽 of the solid of revolution generated by revolving the region bounded by the graphs of 𝒚= 𝟐 𝒙 and 𝒚=𝟑−𝒙 about the 𝒙-axis.
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§7.2: Volume of Disks and Washers

41. §7.2: Volume of Disks and Washers
Example 7
Find the volume 𝑽 of the solid of revolution generated by revolving the region bounded by the graphs of 𝒚= 𝟐 𝒙 and 𝒚=𝟑−𝒙 about the 𝒙-axis.
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§7.2: Volume of Disks and Washers

42. §7.2: Volume of Disks and Washers
Example 7
Find the volume 𝑽 of the solid of revolution generated by revolving the region bounded by the graphs of 𝒚= 𝟐 𝒙 and 𝒚=𝟑−𝒙 about the 𝒙-axis.
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§7.2: Volume of Disks and Washers

43. §7.2: Volume of Disks and Washers
Example 8
Find the volume 𝑽 of the solid of revolution generated by revolving the region bounded by 𝒚= −𝒙 𝟐 +𝟒, 𝒙=𝟏, and 𝒚=𝟎 about the 𝒚-axis.
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§7.2: Volume of Disks and Washers

44. §7.2: Volume of Disks and Washers
Example 8
Find the volume 𝑽 of the solid of revolution generated by revolving the region bounded by 𝒚= −𝒙 𝟐 +𝟒, 𝒙=𝟏, and 𝒚=𝟎 about the 𝒚-axis.
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§7.2: Volume of Disks and Washers

45. §7.2: Volume of Disks and Washers
Example 8
Find the volume 𝑽 of the solid of revolution generated by revolving the region bounded by 𝒚= −𝒙 𝟐 +𝟒, 𝒙=𝟏, and 𝒚=𝟎 about the 𝒚-axis.
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§7.2: Volume of Disks and Washers

46. §7.2: Volume of Disks and Washers
Your Turn
Find the volume 𝑽 of the solid of revolution generated by revolving the region bounded by 𝒚= −𝒙 𝟐 +𝟒, 𝒚=𝟐𝒙+𝟏, and 𝒙=𝟎 about the 𝒙-axis.
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§7.2: Volume of Disks and Washers

47. §7.2: Volume of Disks and Washers
Your Turn
Find the volume 𝑽 of the solid of revolution generated by revolving the region bounded by 𝒚= −𝒙 𝟐 +𝟒, 𝒚=𝟐𝒙+𝟏, and 𝒙=𝟎 about the 𝒙-axis.
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§7.2: Volume of Disks and Washers

48. §7.2: Volume of Disks and Washers
Example 9
Find the volume of the solid of revolution generated by revolving the region bounded by the graphs of 𝒚=𝟐𝒙 and 𝒚= 𝒙 𝟐 about line 𝒚=−𝟓.
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§7.2: Volume of Disks and Washers

49. §7.2: Volume of Disks and Washers
Example 9
Find the volume of the solid of revolution generated by revolving the region bounded by the graphs of 𝒚=𝟐𝒙 and 𝒚= 𝒙 𝟐 about line 𝒚=−𝟓.
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§7.2: Volume of Disks and Washers

50. §7.2: Volume of Disks and Washers
Example 9
Find the volume of the solid of revolution generated by revolving the region bounded by the graphs of 𝒚=𝟐𝒙 and 𝒚= 𝒙 𝟐 about line 𝒚=−𝟓.
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§7.2: Volume of Disks and Washers

51. §7.2: Volume of Disks and Washers
Example 9
Find the volume of the solid of revolution generated by revolving the region bounded by the graphs of 𝒚=𝟐𝒙 and 𝒚= 𝒙 𝟐 about line 𝒚=−𝟓.
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§7.2: Volume of Disks and Washers

52. §7.2: Volume of Disks and Washers
Example 10
Find the volume of the solid of revolution generated by revolving the region bounded by the graphs of 𝒚=𝟐𝒙 and 𝒚= 𝒙 𝟐 about the line 𝒙=𝟐.
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§7.2: Volume of Disks and Washers

53. §7.2: Volume of Disks and Washers
Example 10
Find the volume of the solid of revolution generated by revolving the region bounded by the graphs of 𝒚=𝟐𝒙 and 𝒚= 𝒙 𝟐 about the line 𝒙=𝟐.
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§7.2: Volume of Disks and Washers

54. §7.2: Volume of Disks and Washers
Example 10
Find the volume of the solid of revolution generated by revolving the region bounded by the graphs of 𝒚=𝟐𝒙 and 𝒚= 𝒙 𝟐 about the line 𝒙=𝟐.
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§7.2: Volume of Disks and Washers

55. §7.2: Volume of Disks and Washers
Example 11
Find the volume of a solid formed by revolving 𝒇 𝒙 = 𝒙 𝟐 +𝟐, 𝒈 𝒙 =𝟒− 𝒙 𝟐 about the horizontal line 𝒚=−𝟑.
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§7.2: Volume of Disks and Washers

56. §7.2: Volume of Disks and Washers
Example 11
Find the volume of a solid formed by revolving 𝒇 𝒙 = 𝒙 𝟐 +𝟐, 𝒈 𝒙 =𝟒− 𝒙 𝟐 about the horizontal line 𝒚=−𝟑.
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§7.2: Volume of Disks and Washers

57. §7.2: Volume of Disks and Washers
Example 11
Find the volume of a solid formed by revolving 𝒇 𝒙 = 𝒙 𝟐 +𝟐, 𝒈 𝒙 =𝟒− 𝒙 𝟐 about the horizontal line 𝒚=−𝟑.
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§7.2: Volume of Disks and Washers

58. §7.2: Volume of Disks and Washers
Example 11
Find the volume of a solid formed by revolving 𝒇 𝒙 = 𝒙 𝟐 +𝟐, 𝒈 𝒙 =𝟒− 𝒙 𝟐 about the horizontal line 𝒚=−𝟑.
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§7.2: Volume of Disks and Washers

59. §7.2: Volume of Disks and Washers
Your Turn
Setup the volume of solid by rotating the region of 𝒇 𝒙 =𝟗− 𝒙 𝟐 and 𝒚=𝟏𝟐 for 𝟎,𝟑 about the line 𝒚=𝟏𝟓. Do not solve.
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§7.2: Volume of Disks and Washers

60. AP Multiple Choice Practice Question 1 (non-calculator)
Region 𝑹 is bounded by 𝒇 𝒚 = 𝒚 𝟐 −𝟑 and 𝒈 𝒚 =𝟑𝒚+𝟏, determine the following expressions below that gives the volume when rotated about the line 𝒙=−𝟑. (A) 𝛑 −𝟐 𝟏𝟑 𝟔− 𝒚 𝟐 𝟐 − 𝟐−𝟑𝒚 𝟐 𝒅𝒚 (B) 𝛑 −𝟏 𝟒 𝟐−𝟑𝒚 𝟐 − 𝟔− 𝒚 𝟐 𝟐 𝒅𝒚 (C) 𝛑 −𝟏 𝟒 𝒚 𝟐 −𝟑 𝟐 − 𝟑𝒚+𝟏 𝟐 𝒅𝒚 (D) 𝛑 −𝟏 𝟒 𝟑𝒚+𝟒 𝟐 − 𝒚 𝟐 𝟐 𝒅𝒚
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§7.2: Volume of Disks and Washers

61. AP Multiple Choice Practice Question 1 (non-calculator)
Region 𝑹 is bounded by 𝒇 𝒚 = 𝒚 𝟐 −𝟑 and 𝒈 𝒚 =𝟑𝒚+𝟏, determine the following expressions below that gives the volume when rotated about the line 𝒙=−𝟑.
Vocabulary
Connections and Process
Answer and Justifications
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§7.2: Volume of Disks and Washers

62. §7.2: Volume of Disks and Washers
Assignment
Worksheet
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§7.2: Volume of Disks and Washers