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The equations of the graphs are , and .
(a)
Find the area of the region.
Find the integral limits by equating two curve equations.
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Area of the region bounded by , , and .
Consider and .
The two curves on interval .
The area enclosed by the curves is .
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Solve the integral using integration by parts.
Formula for integration by parts:.
Here and .
Consider .
Apply derivative on each side with respect to .
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Consider .
Apply integral on each side.
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Substitute corresponding values in .
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The area of the region 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.
Formula for the volume of the solid with the Washer method,
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The outer radius of revolution is .
The inner radius of revolution is .
Solve the integral using integration by parts.
Formula for integration by parts:.
Here and .
Consider .
Apply derivative on each side with respect to .
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Consider .
Apply integral on each side.
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Substitute corresponding values in .
Again apply integration by parts.
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The volume of the solid generated by revolving the region about the -axis is .
(c)
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
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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 .
(d)
Find the centroid of the region.
Moments and center of mass of a planar lamina:
Let and be continuous functions such that f on , and consider the planar lamina of uniform density bounded by the graphs of and .
The moments about the -and -axis are ..
The center of mass is and , where is the mass of the lamina.
Substitute , , and in .
Substitute .
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Substitute , , and in .
Substitute .
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The centroid of the region is .
(a) The area of the region is .
(b) The volume of the solid generated by revolving the region about the -axis is .
(c) The volume of the solid generated by revolving the region about the -axis is .
(d) The centroid of the region is .
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