solids of revolution
DESCRIPTION
Solids of Revolution. 4-G. Disk method. Revolve about a horizontal axis Slice perpendicular to axis – slices vertical Integrate in terms of x. Revolve about a vertical axis Slice perpendicular to axis – slices horizontal Integrate in terms of y. Find Volume – Disk Method. Washer Method. - PowerPoint PPT PresentationTRANSCRIPT
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SOLIDS OF REVOLUTION
4-G
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Disk method
![Page 3: Solids of Revolution](https://reader036.vdocuments.mx/reader036/viewer/2022062520/56816384550346895dd46b2e/html5/thumbnails/3.jpg)
Find Volume – Disk Method
• Revolve about a horizontal axis• Slice perpendicular to axis – slices vertical• Integrate in terms of x
• Revolve about a vertical axis
• Slice perpendicular to axis – slices horizontal
• Integrate in terms of y
b
a
dxxfV 2))(( d
c
dyyfV 2))((
![Page 4: Solids of Revolution](https://reader036.vdocuments.mx/reader036/viewer/2022062520/56816384550346895dd46b2e/html5/thumbnails/4.jpg)
Washer Method
![Page 5: Solids of Revolution](https://reader036.vdocuments.mx/reader036/viewer/2022062520/56816384550346895dd46b2e/html5/thumbnails/5.jpg)
Volume of Washer:
Horizontal axis of revolution
• Slice with rectangle perpendicular to axis
• Integrate in terms of x
Vertical axis of revolution• Slice with rectangles
perpendicular to axis• Integrate in terms of y
hrRV 22
b
a
dxxgxfV 22 )()((
d
c
dyygyfV 22 )()((
Find Volume – Washer Method
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1) Find the volume of the solid is bounded by the curve y = x and the x-axis between x = 0 and x = 1 rotated about the x-axis
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2) Find the volume of the solid when the region bounded by
is revolved about the x-axis4 x to2 xfrom and 23 xyxy
![Page 8: Solids of Revolution](https://reader036.vdocuments.mx/reader036/viewer/2022062520/56816384550346895dd46b2e/html5/thumbnails/8.jpg)
3) Find the volume of the solid generated when the region bounded by
is revolved about the x-axisaxis- x theand 9 x line theand xy
![Page 9: Solids of Revolution](https://reader036.vdocuments.mx/reader036/viewer/2022062520/56816384550346895dd46b2e/html5/thumbnails/9.jpg)
4) Find the volume of the solid generated when the region bounded by
is revolved about the line y = 4 4y line theand 2 xy
![Page 10: Solids of Revolution](https://reader036.vdocuments.mx/reader036/viewer/2022062520/56816384550346895dd46b2e/html5/thumbnails/10.jpg)
5) Set up the integrals for finding the volume of the solid formed by revolving the region formed by
a) About the x-axis b) about y = -1 02 yxxy and
x y
R r
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6) Find the volume of the solid generated when the region bounded by
is revolved about the line y = 36x y line theand 26 2 xxy
![Page 12: Solids of Revolution](https://reader036.vdocuments.mx/reader036/viewer/2022062520/56816384550346895dd46b2e/html5/thumbnails/12.jpg)
7) Find the volume of the solid generated when the region bounded by
is revolved about the y-axis4,02 yxxy and ,
![Page 13: Solids of Revolution](https://reader036.vdocuments.mx/reader036/viewer/2022062520/56816384550346895dd46b2e/html5/thumbnails/13.jpg)
HOME WORKFinding Volumes of solids using washers and disks worksheet