The equations of the graphs are , , and .
(a)
Find the volume of the solid generated by revolving the region about the -axis.
The volume of the solid generated revolving about the - axis.
Formula for the volume of the solid with the Washer method,
.
The outer radius of revolution is .
The inner radius of revolution is .
.
Consider .
Solve the integral using integration by parts.
Formula for integration by parts:.
Here and .
Consider .
Apply derivative on each side with respect to .
.
Consider .
Apply integral on each side.
.
Substitute corresponding values in .
.
Substitute in .
The volume of the solid generated by revolving the region about the -axis is .
(b)
Find the volume of the solid generated by revolving the region about the -axis.
The volume of the solid generated revolving about the - axis is .
Here and .
.
Solve the integral using integration by parts.
Formula for integration by parts:.
Here and .
Consider .
Apply derivative on each side with respect to .
.
Consider .
Apply integral on each side.
.
Substitute corresponding values in .
The volume of the solid generated by revolving the region about the -axis is .
(c)
Find the centroid of the region.
Moments and center of mass of a planar lamina:
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 -and -axes are ..
The center of mass is and , where is the mass of the lamina.
Find .
Substitute , and in .
.
Find .
Substitute , and in .
Substitute .
.
Find .
Substitute , and in .
.
Substitute .
.
Find the centroid.
Substitute in .
.
Substitute in .
.
The centroid of the region is .
(a) The volume of the solid generated by revolving the region about the -axis is .
(b) The volume of the solid generated by revolving the region about the -axis is .
(c) The centroid of the region is .
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