$\"\"$

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

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Find the intersection points, by equating curves.

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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 curves bounded between $\"\"$ to $\"\"$ .

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Cylindrical shells method:

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The volume of the solid obtained by rotating about the $\"\"$ -axis the region under the curve is

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$\"\"$

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$\"\"$ .

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Integrate between $\"\"$ to $\"\"$ .

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

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The shell height is $\"\"$ .

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$\"\"$

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

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$\"\"$

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$\"\"$ .

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Consider $\"\"$ .

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Apply parts of integration formula: $\"\"$ .

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Here $\"\"$ and $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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Integrate on each side.

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$\"\"$

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$\"\"$ .

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Substitute the corresponding values in $\"\"$ .

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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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Consider $\"\"$ .

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Apply parts of integration formula: $\"\"$ .

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Here $\"\"$ and $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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Integrate on each side.

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$\"\"$

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$\"\"$ .

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Substitute the corresponding values in $\"\"$ .

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

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ cubic units.

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$\"\"$

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$\"\"$ cubic units.