Volumes of Revolution Disks and Washers Lesson 7.2.
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Transcript of Volumes of Revolution Disks and Washers Lesson 7.2.
![Page 1: Volumes of Revolution Disks and Washers Lesson 7.2.](https://reader035.fdocuments.us/reader035/viewer/2022072016/56649edb5503460f94bebd4a/html5/thumbnails/1.jpg)
Volumes of RevolutionDisks and Washers
Lesson 7.2
![Page 2: Volumes of Revolution Disks and Washers Lesson 7.2.](https://reader035.fdocuments.us/reader035/viewer/2022072016/56649edb5503460f94bebd4a/html5/thumbnails/2.jpg)
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Revolving a Function
• Consider a function f(x) on the interval [a, b]
• Now consider revolvingthat segment of curve about the x axis
• What kind of functions generated these solids of revolution?
f(x)
a b
![Page 3: Volumes of Revolution Disks and Washers Lesson 7.2.](https://reader035.fdocuments.us/reader035/viewer/2022072016/56649edb5503460f94bebd4a/html5/thumbnails/3.jpg)
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Disks
• We seek ways of usingintegrals to determine thevolume of these solids
• Consider a disk which is a slice of the solid What is the radius What is the thickness What then, is its volume?
dx
f(x)
2Volume of slice = ( )f x dx
![Page 4: Volumes of Revolution Disks and Washers Lesson 7.2.](https://reader035.fdocuments.us/reader035/viewer/2022072016/56649edb5503460f94bebd4a/html5/thumbnails/4.jpg)
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Disks
• To find the volume of the whole solid we sum thevolumes of the disks
• Shown as a definite integral
f(x)
a b
2( )
b
a
V f x dx
![Page 5: Volumes of Revolution Disks and Washers Lesson 7.2.](https://reader035.fdocuments.us/reader035/viewer/2022072016/56649edb5503460f94bebd4a/html5/thumbnails/5.jpg)
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Try It Out!
• Try the function y = x3 on the interval 0 < x < 2 rotated about x-axis
![Page 6: Volumes of Revolution Disks and Washers Lesson 7.2.](https://reader035.fdocuments.us/reader035/viewer/2022072016/56649edb5503460f94bebd4a/html5/thumbnails/6.jpg)
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Washers• Consider the area
between two functions rotated about the axis
• Now we have a hollow solid
• We will sum the volumes of washers
• As an integral
f(x)
a b
g(x)
2 2( ) ( )
b
a
V f x g x dx
![Page 7: Volumes of Revolution Disks and Washers Lesson 7.2.](https://reader035.fdocuments.us/reader035/viewer/2022072016/56649edb5503460f94bebd4a/html5/thumbnails/7.jpg)
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Application
• Given two functions y = x2, and y = x3
Revolve region between about x-axis
What will be the limits of
integration?
What will be the limits of
integration?
1
2 22 3
0
V x x dx
![Page 8: Volumes of Revolution Disks and Washers Lesson 7.2.](https://reader035.fdocuments.us/reader035/viewer/2022072016/56649edb5503460f94bebd4a/html5/thumbnails/8.jpg)
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Revolving About y-Axis
• Also possible to revolve a function about the y-axis Make a disk or a washer to be horizontal
• Consider revolving a parabola about the y-axis How to represent the
radius? What is the thickness
of the disk?
![Page 9: Volumes of Revolution Disks and Washers Lesson 7.2.](https://reader035.fdocuments.us/reader035/viewer/2022072016/56649edb5503460f94bebd4a/html5/thumbnails/9.jpg)
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Revolving About y-Axis
• Must consider curve asx = f(y) Radius = f(y) Slice is dy thick
• Volume of the solid rotatedabout y-axis
2( )
b
a
V f y dy
![Page 10: Volumes of Revolution Disks and Washers Lesson 7.2.](https://reader035.fdocuments.us/reader035/viewer/2022072016/56649edb5503460f94bebd4a/html5/thumbnails/10.jpg)
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Flat Washer
• Determine the volume of the solid generated by the region between y = x2 and y = 4x, revolved about the y-axis Radius of inner circle?
• f(y) = y/4
Radius of outer circle?•
Limits?• 0 < y < 16
( )f y y
![Page 11: Volumes of Revolution Disks and Washers Lesson 7.2.](https://reader035.fdocuments.us/reader035/viewer/2022072016/56649edb5503460f94bebd4a/html5/thumbnails/11.jpg)
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Assignment
• Lesson 7.2
• Page 272
• Exercises 1 – 23 odd