$\"\"$

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

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(a)

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Find volume generated by rotating the region about $\"\"$ -axis.

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Method of washer for volume of solid:

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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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$\"\"$

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Definition of logarithm : $\"\"$ if and only if $\"\"$ .

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$\"\"$

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

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$\"\"$ .

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Inner radius : $\"\"$ .

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Outer radius : $\"\"$ .

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Integral limits are $\"\"$ and $\"\"$ .

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$\"\"$

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Volume of the solid obtained by rotating $\"\"$ and $\"\"$ is

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$\"\"$

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$\"\"$

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Volume generated by rotating the region about $\"\"$ -axis is $\"\"$ .

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$\"\"$

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(b)

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Find volume generated by rotating the region about $\"\"$ -axis.

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Method of Cylinders :

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The volume of the solid obtained by rotating about $\"\"$ -axis, the region of the curve from $\"\"$ to $\"\"$ is $\"\"$ .

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

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$\"\"$

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Consider $\"\"$ .

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Integration by parts $\"\"$ .

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Let $\"\"$ then $\"\"$ .

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Let $\"\"$ then $\"\"$ .

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Substitute $\"\"$ , $\"\"$ , $\"\"$ , $\"\"$ in the above equation.

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$\"\"$

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$\"\"$ .

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$\"\"$

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Substitute $\"\"$ in equation (1).

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$\"\"$

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Volume of the solid generated by rotating the region about $\"\"$ -axis is $\"\"$ .

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$\"\"$

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Volume generated by rotating the region about $\"\"$ -axis is $\"\"$ .

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Volume of the solid generated by rotating the region about $\"\"$ -axis is $\"\"$ .