$\"\"$

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The curve equations are $\"\"$ .

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Shell method:

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The plane region is revolving about the $\"\"$ -axis.

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

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Observe the figure:

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The distance from the center of the rectangle to the axis of revolution is $\"\"$ .

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The height of the rectangle is $\"\"$ .

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Substitute $\"\"$ .

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$\"\"$

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$\"\"$ .

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The integrating limits are $\"\"$ to $\"\"$ .

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

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$\"\"$ .

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$\"\"$

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Evaluate the integral $\"\"$ .

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$\"\"$

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Apply formula: $\"\"$ .

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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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$\"\"$ .

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$\"\"$

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$\"\"$ .