$\"\"$

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The equations are $\"\"$ , $\"\"$ , $\"\"$ and $\"\"$ .

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Method of disk :

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The volume of the solid $\"\"$ is $\"\"$ , where $\"\"$ is the cross sectional area of the solid $\"\"$ .

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$\"\"$ .

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Radius $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ cubic units.

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Volume of the solid is $\"\"$ cubic units.

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$\"\"$

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The equations are $\"\"$ , $\"\"$ , $\"\"$ and $\"\"$ .

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Centroid of the region is

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$\"\"$

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Area $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ sq-units.

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$\"\"$

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$\"\"$

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Find the integral using integration by parts.

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Integration by parts : $\"\"$ .

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Consider $\"\"$ and $\"\"$ .

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Consider $\"\"$ .

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Apply derivative on each side with respect to $\"\"$ .

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$\"\"$ .

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Consider $\"\"$ .

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Apply integral on each side.

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$\"\"$

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$\"\"$ .

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Substitute corresponding values in $\"\"$ .

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$\"\"$

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$\"\"$

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Find $\"\"$ co-ordinate of the centroid.

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$\"\"$

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Centriod $\"\"$ .

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$\"\"$

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Volume of the solid is $\"\"$ cubic units.

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Centriod $\"\"$ .