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| PAGE: 550 | SET: Exercises | PROBLEM: 47 |
The equations are 
, 
, 
and 
.
Find the volume of the solid generated by revolving the region about the 
-axis.
The Volume of the solid solid generated by revolving the region about the 
-axis, for the function 
in the interval 
is 
.
Here 
and 
.
The volume of the solid is 
.
Apply formula : 
.
The volume of the solid is 
.
The center of mass:
Let 
and 
be continuous functions such that 
on 
, and consider the planar lamina of uniform density 
bounded by the graphs of 
and 
.
The moments about the -axis and 
-axis are
.
.
The center of mass 
is 
and 
, where 
is the mass of the lamina.
Find the centroid of the region.
Here 
, 
and 
.
Find 
.
Substitute
, 
and 
in
.
Substitute 
in 
.
.
Find 
.
Substitute, and in .
.
Find the center of mass .
The center of mass is and , where is the mass of the lamina.
Substitute, and in .
Apply formula : 
.
.
Substitute 
and in 
in .
.
Substitute 
and .
The centroid of the solid is 
.
The volume of the solid is and centroid is .
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