$\"\"$

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Therom of Pappus :

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Let $\"\"$ be a region in a plane and and let $\"\"$ be the same plane such that L does not intersect with the interior of $\"\"$ .

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If $\"\"$ is the distance between centeriod $\"\"$ and the line then the volume $\"\"$ of the solid of the revolution formed by revolving $\"\"$ about the line is $\"\"$ .

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$\"\"$

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The region is $\"\"$ , $\"\"$ and $\"\"$ .

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Graph:

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Graph the equations $\"\"$ , $\"\"$ and $\"\"$ .

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$\"\"$

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Observe the graph:

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The area bounded by the three equations is $\"\"$ .

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Substitute $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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The distance between the centriod and the $\"\"$ -axis is $\"\"$ .

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Here $\"\"$ .

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Substitute $\"\"$ and $\"\"$ .

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$\"\"$ .

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$\"\"$

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Consider $\"\"$ .

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$\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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The volume of the solid of the revolution is $\"\"$ .

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Substitute $\"\"$ and $\"\"$ .

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$\"\"$

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$\"\"$ .

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$\"\"$

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$\"\"$ .