Volumes Using Cylindrical Shells
J. Gonzalez-Zugasti, University of Massachusetts - Lowell 1
Volume of a Shell
Volume of outer cylinderβ Volume of inner cylinder
= ππ22β β ππ12β = πβ π22 β π12 = πβ π2 + π1 π2 β π1 = 2π
π2 + π12
β π2 β π1 = 2π(average radius) height thickness
J. Gonzalez-Zugasti, University of Massachusetts - Lowell 2
Revolve π¦ = π π₯ about the π¦-axis
J. Gonzalez-Zugasti, University of Massachusetts - Lowell 3
Revolve π¦ = π π₯ about the π¦-axis
On each slice: average radius
=π₯π + π₯πβ1
2= ππ
height = π ππ thickness = βπ₯π
ππ β 2ππππ ππ βπ₯π
J. Gonzalez-Zugasti, University of Massachusetts - Lowell 4
Method of Cylindrical Shells
π = οΏ½ππ
π
π=1
β οΏ½ 2ππππ ππ βπ₯π
π
π=1
π = οΏ½ 2ππ₯π π₯π
πππ₯
π = οΏ½ 2π radius heightπ
πππ₯
J. Gonzalez-Zugasti, University of Massachusetts - Lowell 5
Surface area of a cylinder
Revolve π¦ = π π₯ about π₯ = πΏ
π
= οΏ½ 2π radius heightπ
πππ₯
= οΏ½ 2π π₯ β πΏ π π₯π
πππ₯
J. Gonzalez-Zugasti, University of Massachusetts - Lowell 6
Example 1
Use cylindrical shells to find the volume of the solid generated when the region enclosed between π¦ = π₯, π₯ = 1, π₯ = 4 and the π₯-axis is revolved about the π¦-axis. Solution: Sketch the curve.
J. Gonzalez-Zugasti, University of Massachusetts - Lowell 7
π¦ = π₯
Example 1 (continued)
π
= οΏ½ 2π radius heightπ
πππ₯
= οΏ½ 2ππ₯ π₯4
1ππ₯
= οΏ½ 2ππ₯3 2β4
1ππ₯
= β― =124π
5
J. Gonzalez-Zugasti, University of Massachusetts - Lowell 8
π¦ = π₯
Example 2 Redo Example 1 using the Washer Method. (Find the volume of the solid generated when the region enclosed between π¦ = π₯, π₯ = 1, π₯ = 4 and the π₯-axis is revolved about the π¦-axis.) Solution: Sketch the curve.
J. Gonzalez-Zugasti, University of Massachusetts - Lowell 9
π¦ = π₯ βΉ π₯ = π¦2
Example 2 (continued)
π = οΏ½ π 42 β 121
0ππ¦
+ οΏ½ π 42 β π¦2 22
1ππ¦
= β― =124π
5
J. Gonzalez-Zugasti, University of Massachusetts - Lowell 10
π¦ = π₯ βΉ π₯ = π¦2
Example 3 Use cylindrical shells to find the volume of the solid generated when the region π in the first quadrant enclosed between π¦ = π₯ and π¦ = π₯2 is revolved about the π¦-axis. Solution: Sketch region to be revolved.
J. Gonzalez-Zugasti, University of Massachusetts - Lowell 11
Example 3 (continued)
π
= οΏ½ 2π radius heightπ
πππ₯
= οΏ½ 2π π₯ π₯ β π₯21
0ππ₯
= β― =π6
J. Gonzalez-Zugasti, University of Massachusetts - Lowell 12
Example 4 (Revolve π₯ = π’ π¦ about the π₯-axis)
Use cylindrical shells to find the volume of the solid generated when the region π under π¦ = π₯2 over the interval 0,2 is revolved about the π₯-axis. Solution: Sketch region to be revolved.
J. Gonzalez-Zugasti, University of Massachusetts - Lowell 13
Example 4 (continued)
π
= οΏ½ 2π radius heightπ
πππ¦
= οΏ½ 2ππ¦ 2 β π¦4
0ππ¦
= β― =32π
5
J. Gonzalez-Zugasti, University of Massachusetts - Lowell 14
J. Gonzalez-Zugasti, University of Massachusetts - Lowell 15
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