section 6.3: volumes by cylindrical shells •objective · 2007-04-20 · 6.3 16 typed solution the...
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![Page 1: Section 6.3: Volumes By Cylindrical Shells •Objective · 2007-04-20 · 6.3 16 Typed Solution The region bounded by the parabola and the line y=2 is rotated about the y axis. Find](https://reader034.vdocument.in/reader034/viewer/2022042302/5ecd1e5841b2ea278b4a8cf8/html5/thumbnails/1.jpg)
6.3 1
Section 6.3: Volumes By Cylindrical Shells
• Objective– Understand how to find the volume of a solid of
revolution using the method of cylindrical shells
![Page 2: Section 6.3: Volumes By Cylindrical Shells •Objective · 2007-04-20 · 6.3 16 Typed Solution The region bounded by the parabola and the line y=2 is rotated about the y axis. Find](https://reader034.vdocument.in/reader034/viewer/2022042302/5ecd1e5841b2ea278b4a8cf8/html5/thumbnails/2.jpg)
6.3 2
Problem
2 32y x x= −
Find the volume of the solid obtained by rotating about the y axis the region bounded by
2 32 0y x x and y= − =
![Page 3: Section 6.3: Volumes By Cylindrical Shells •Objective · 2007-04-20 · 6.3 16 Typed Solution The region bounded by the parabola and the line y=2 is rotated about the y axis. Find](https://reader034.vdocument.in/reader034/viewer/2022042302/5ecd1e5841b2ea278b4a8cf8/html5/thumbnails/3.jpg)
6.3 3
Discussion
2 32y x x= −
![Page 4: Section 6.3: Volumes By Cylindrical Shells •Objective · 2007-04-20 · 6.3 16 Typed Solution The region bounded by the parabola and the line y=2 is rotated about the y axis. Find](https://reader034.vdocument.in/reader034/viewer/2022042302/5ecd1e5841b2ea278b4a8cf8/html5/thumbnails/4.jpg)
6.3 4
Discussion
2 32y x x= −
If we tried to use the washer method, we would have to solve for x in terms of y. This is not possible with the tools we have. There is a way out.
![Page 5: Section 6.3: Volumes By Cylindrical Shells •Objective · 2007-04-20 · 6.3 16 Typed Solution The region bounded by the parabola and the line y=2 is rotated about the y axis. Find](https://reader034.vdocument.in/reader034/viewer/2022042302/5ecd1e5841b2ea278b4a8cf8/html5/thumbnails/5.jpg)
6.3 5
The Shell Methody
a b
y=f(x)
R
x
![Page 6: Section 6.3: Volumes By Cylindrical Shells •Objective · 2007-04-20 · 6.3 16 Typed Solution The region bounded by the parabola and the line y=2 is rotated about the y axis. Find](https://reader034.vdocument.in/reader034/viewer/2022042302/5ecd1e5841b2ea278b4a8cf8/html5/thumbnails/6.jpg)
6.3 6
The Shell Method
y
a b
y=f(x)
R
x
Given a continuous nonnegative function f defined on [a,b]. Consider the region R. Revolve R about the y axis. A solid called a cylindrical shell is generated. What is its volume?
![Page 7: Section 6.3: Volumes By Cylindrical Shells •Objective · 2007-04-20 · 6.3 16 Typed Solution The region bounded by the parabola and the line y=2 is rotated about the y axis. Find](https://reader034.vdocument.in/reader034/viewer/2022042302/5ecd1e5841b2ea278b4a8cf8/html5/thumbnails/7.jpg)
6.3 7
Shell Method: General Formula
2 ( ) ( )b
a
V p x h x dxπ= ∫ 2 ( ) ( )d
c
V p y h y dyπ= ∫
p = distance from the center of the
rectangle to the axis of revolution
h =height of the rectangle
![Page 8: Section 6.3: Volumes By Cylindrical Shells •Objective · 2007-04-20 · 6.3 16 Typed Solution The region bounded by the parabola and the line y=2 is rotated about the y axis. Find](https://reader034.vdocument.in/reader034/viewer/2022042302/5ecd1e5841b2ea278b4a8cf8/html5/thumbnails/8.jpg)
6.3 8
Volume of solid: Shell Method Formula
2 ( )b
a
V x f x dxπ= ∫ Rectangular slice in region rotated is parallel to the y axis(axis of rotation)
2 ( )d
c
V y g y dyπ= ∫Rectangular slice in region rotated is parallel to the x axis(axis of rotation)
Typical shell has radius x, circumference 2 ( )x and height f xπ
![Page 9: Section 6.3: Volumes By Cylindrical Shells •Objective · 2007-04-20 · 6.3 16 Typed Solution The region bounded by the parabola and the line y=2 is rotated about the y axis. Find](https://reader034.vdocument.in/reader034/viewer/2022042302/5ecd1e5841b2ea278b4a8cf8/html5/thumbnails/9.jpg)
6.3 9
Steps to find volume using the Shell Method
(a) Sketch the region R
(b) Show a typical rectangular slice properly labeled
(c) Write a formula for the approximate volume of the
shell generated by this slice
(d) Set up the corresponding integral
(e) Evaluate the integral
![Page 10: Section 6.3: Volumes By Cylindrical Shells •Objective · 2007-04-20 · 6.3 16 Typed Solution The region bounded by the parabola and the line y=2 is rotated about the y axis. Find](https://reader034.vdocument.in/reader034/viewer/2022042302/5ecd1e5841b2ea278b4a8cf8/html5/thumbnails/10.jpg)
6.3 10
Example
Find the volume of the solid of revolution formed by revolving the region bounded by the graph of and the y axis
about the x-axis.
2yx e−= 0 1y≤ ≤
![Page 11: Section 6.3: Volumes By Cylindrical Shells •Objective · 2007-04-20 · 6.3 16 Typed Solution The region bounded by the parabola and the line y=2 is rotated about the y axis. Find](https://reader034.vdocument.in/reader034/viewer/2022042302/5ecd1e5841b2ea278b4a8cf8/html5/thumbnails/11.jpg)
6.3 11
![Page 12: Section 6.3: Volumes By Cylindrical Shells •Objective · 2007-04-20 · 6.3 16 Typed Solution The region bounded by the parabola and the line y=2 is rotated about the y axis. Find](https://reader034.vdocument.in/reader034/viewer/2022042302/5ecd1e5841b2ea278b4a8cf8/html5/thumbnails/12.jpg)
6.3 12
Continued 2
( ) yh y e−=
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6.3 13
![Page 14: Section 6.3: Volumes By Cylindrical Shells •Objective · 2007-04-20 · 6.3 16 Typed Solution The region bounded by the parabola and the line y=2 is rotated about the y axis. Find](https://reader034.vdocument.in/reader034/viewer/2022042302/5ecd1e5841b2ea278b4a8cf8/html5/thumbnails/14.jpg)
6.3 14
Example
The region bounded by the parabola and the
line y=2 is rotated about the y axis. Find the volume of the
resulting solid.
23y x x= −
![Page 15: Section 6.3: Volumes By Cylindrical Shells •Objective · 2007-04-20 · 6.3 16 Typed Solution The region bounded by the parabola and the line y=2 is rotated about the y axis. Find](https://reader034.vdocument.in/reader034/viewer/2022042302/5ecd1e5841b2ea278b4a8cf8/html5/thumbnails/15.jpg)
6.3 15
Solution
![Page 16: Section 6.3: Volumes By Cylindrical Shells •Objective · 2007-04-20 · 6.3 16 Typed Solution The region bounded by the parabola and the line y=2 is rotated about the y axis. Find](https://reader034.vdocument.in/reader034/viewer/2022042302/5ecd1e5841b2ea278b4a8cf8/html5/thumbnails/16.jpg)
6.3 16
Typed Solution
The region bounded by the parabola and the
line y=2 is rotated about the y axis. Find the volume of the
resulting solid.
23y x x= −
The length of a rectangular slice is ( )23 2h x x= − −
and the distance from middle of the rectangle to the y axis is x
![Page 17: Section 6.3: Volumes By Cylindrical Shells •Objective · 2007-04-20 · 6.3 16 Typed Solution The region bounded by the parabola and the line y=2 is rotated about the y axis. Find](https://reader034.vdocument.in/reader034/viewer/2022042302/5ecd1e5841b2ea278b4a8cf8/html5/thumbnails/17.jpg)
6.3 17
Solution
( )2
2
1
2 3 2V x x x dxπ= − −∫ ( )2
2 3
1
2 3 22
x x x dx ππ= − − =∫
![Page 18: Section 6.3: Volumes By Cylindrical Shells •Objective · 2007-04-20 · 6.3 16 Typed Solution The region bounded by the parabola and the line y=2 is rotated about the y axis. Find](https://reader034.vdocument.in/reader034/viewer/2022042302/5ecd1e5841b2ea278b4a8cf8/html5/thumbnails/18.jpg)
6.3 18
Example: Shell Method; Rotation about line parallel to x –axis.
The region bounded by the line y=1-x, the x-axis, and the y-axis is revolved about the line y=-1.
Find the volume of the solid generated by the (a) shell method (b) washer method
![Page 19: Section 6.3: Volumes By Cylindrical Shells •Objective · 2007-04-20 · 6.3 16 Typed Solution The region bounded by the parabola and the line y=2 is rotated about the y axis. Find](https://reader034.vdocument.in/reader034/viewer/2022042302/5ecd1e5841b2ea278b4a8cf8/html5/thumbnails/19.jpg)
6.3 19
![Page 20: Section 6.3: Volumes By Cylindrical Shells •Objective · 2007-04-20 · 6.3 16 Typed Solution The region bounded by the parabola and the line y=2 is rotated about the y axis. Find](https://reader034.vdocument.in/reader034/viewer/2022042302/5ecd1e5841b2ea278b4a8cf8/html5/thumbnails/20.jpg)
6.3 20
![Page 21: Section 6.3: Volumes By Cylindrical Shells •Objective · 2007-04-20 · 6.3 16 Typed Solution The region bounded by the parabola and the line y=2 is rotated about the y axis. Find](https://reader034.vdocument.in/reader034/viewer/2022042302/5ecd1e5841b2ea278b4a8cf8/html5/thumbnails/21.jpg)
6.3 21
Typed: Solution: Shell Method; Rotation about line parallel to x –axis.
( ) ( )1
0
2 1 1V y y dyπ= + −∫ ( )1
2
0
2 1 y dyπ= −∫13
0
423 3yyπ π
⎛ ⎞= − =⎜ ⎟
⎝ ⎠
![Page 22: Section 6.3: Volumes By Cylindrical Shells •Objective · 2007-04-20 · 6.3 16 Typed Solution The region bounded by the parabola and the line y=2 is rotated about the y axis. Find](https://reader034.vdocument.in/reader034/viewer/2022042302/5ecd1e5841b2ea278b4a8cf8/html5/thumbnails/22.jpg)
6.3 22
Solution: Washer Method; Rotation about line parallel to x –axis.
( ) ( )1
2 2
0
( ) ( )V f x g x dxπ ⎡ ⎤= −⎣ ⎦∫
( ) ( )1
2 2
0
1 ( 1) 0 ( 1)x dxπ= − − − − − −⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦∫
( )1
2
0
2 1x dxπ ⎡ ⎤− −⎣ ⎦∫43π
=