volumes by cylindrical shells shell method.pdfcylindrical shells. then . . . circumference of the...
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![Page 1: VOLUMES BY CYLINDRICAL SHELLS Shell Method.pdfcylindrical shells. Then . . . circumference of the base circle with radius x a solid of revolution surface area of the cylindrical shell](https://reader035.vdocuments.us/reader035/viewer/2022070214/6111d6b81cdac579a92b9e7f/html5/thumbnails/1.jpg)
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VOLUMES BY CYLINDRICAL SHELLS
In the disk method, the axis of revolution must be adjacent to the region being rotated and is the axis of the independent variable; in the method of cylindrical shells, the axis of revolution might be separated from the region being rotated and is the axis of the dependent variable.
USE DISKS USE SHELLS
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BASIC DIFFERENCE IN CONCEPT:
Disk method divides the solid into infinitesimal flat cross‐sectional disks.
Shell method divides the solid into infinitesimal curved cylindrical shells.
Then . . .
circumference of the base circle with radius x
a solid of revolution
surface area of the cylindrical shell at x (since, if it is cut open and rolled out flat, it is a rectangle of length and width
AREA =
approximate volume of the “infinitesimal cylindrical shell” at x. Then “add up” all of the infinitesimal volumes to get the volume, V, of the solid:
VERTICAL AXIS OF REVOLUTION
HORIZONTAL AXIS OF REVOLUTION
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EX #1: Find the volume of the solid of revolution obtained by rotating the region bounded by the curve
and the lines andabout the y‐axis.
![Page 4: VOLUMES BY CYLINDRICAL SHELLS Shell Method.pdfcylindrical shells. Then . . . circumference of the base circle with radius x a solid of revolution surface area of the cylindrical shell](https://reader035.vdocuments.us/reader035/viewer/2022070214/6111d6b81cdac579a92b9e7f/html5/thumbnails/4.jpg)
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EX #2: Find the volume of the solid of revolution obtained by rotating the region enclosed between
and the line about the y‐axis.
![Page 5: VOLUMES BY CYLINDRICAL SHELLS Shell Method.pdfcylindrical shells. Then . . . circumference of the base circle with radius x a solid of revolution surface area of the cylindrical shell](https://reader035.vdocuments.us/reader035/viewer/2022070214/6111d6b81cdac579a92b9e7f/html5/thumbnails/5.jpg)
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EX #3: Find the volume of the solid of revolution obtained by rotating the region bounded by
and the x‐axis about the y‐axis.
![Page 6: VOLUMES BY CYLINDRICAL SHELLS Shell Method.pdfcylindrical shells. Then . . . circumference of the base circle with radius x a solid of revolution surface area of the cylindrical shell](https://reader035.vdocuments.us/reader035/viewer/2022070214/6111d6b81cdac579a92b9e7f/html5/thumbnails/6.jpg)
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EX #4: Find the volume of the solid of revolution obtained by rotating the region bounded by the graphs of
andabout the line
![Page 7: VOLUMES BY CYLINDRICAL SHELLS Shell Method.pdfcylindrical shells. Then . . . circumference of the base circle with radius x a solid of revolution surface area of the cylindrical shell](https://reader035.vdocuments.us/reader035/viewer/2022070214/6111d6b81cdac579a92b9e7f/html5/thumbnails/7.jpg)
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EX #5: Find the volume of the solid formed by revolving the region bounded by the graphs of
and about the x‐axis.
![Page 8: VOLUMES BY CYLINDRICAL SHELLS Shell Method.pdfcylindrical shells. Then . . . circumference of the base circle with radius x a solid of revolution surface area of the cylindrical shell](https://reader035.vdocuments.us/reader035/viewer/2022070214/6111d6b81cdac579a92b9e7f/html5/thumbnails/8.jpg)
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EX #6: Find the volume of the solid formed by revolving the region bounded by the graphs of and about the x‐axis.