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30B Volume Solids
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Volume of Solids
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The volume of a solid right prism or cylinder is the area of the base times the height.
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How would we find the volume of a shape like this?
Disk Method
Volume of one penny:
Volume of a stack of pennies:
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EX 1 Find the volume of the solid of revolution obtained by revolving the region
bounded by , the x-axis and the line x=9 about the x-axis.
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EX 2 Find the volume of the solid generated by revolving the region
enclosed by and about the y-axis.
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How would we find the volume of a washer?Washer Method
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EX 3 Find the volume of the solid generated by revolving about the x-axis the
region bounded by and .
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EX 4 Find the volume of the solid generated by revolving about the line y = 2
the region in the first quadrant bounded by these parabolas and the y-axis.
(Hint: Always measure radius from the axis of revolution.)
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How would we find the volume of a label we peel off a can?Shell Method
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EX 5 Find the volume of the solid generated when the region bounded by
these three equations is revolved about the y-axis.
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EX 6 Find the volume of the solid generated when the region bounded by these
equations is revolved about the y-axis in two ways.
① Shell method
② Disk method
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EX 7 Find the volume of the solid generated when the region in quadrant 1 bounded by these equations is revolved about the line x = 3.
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EX 8 A region, R is shown below. Set up an integral for the volume obtained by revolving R about the given line.
a) The y-axis
b) The x-axis
c) The line y = 3
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