$\"\"$

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The ellipse equation is $\"\"$ where $\"\"$ .

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(a) Find the surface area of the ellipse rotated about its major axis.

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So the major axis is along $\"\"$ -axis and has length $\"\"$ .

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The surface area of the soild obtained after rotation of the $\"\"$ -axis is $\"\"$ .

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The parametric equations of the ellipse are $\"\"$ and $\"\"$ , $\"\"$ .

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$\"\"$

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$\"\"$ .

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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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$\"\"$ .

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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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$\"\"$ .

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

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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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Consider $\"\"$ .

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

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$\"\"$

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$\"\"$

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$\"\"$ .

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

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$\"\"$

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$\"\"$

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Substitute $\"\"$ .

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$\"\"$

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Apply the 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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The surface area of the ellipse rotated about its major axis is $\"\"$ .

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$\"\"$

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(b) Find the surface area of the ellipse rotated about its minor axis.

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The ellipse equation is $\"\"$ where $\"\"$ .

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So that the major axis is along $\"\"$ -axis and has length $\"\"$ .

\

The surface area of the soild obtained after rotation of the $\"\"$ -axis is $\"\"$ .

\

The parametric equations of the ellipse are $\"\"$ and $\"\"$ , $\"\"$ . \ \

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$\"\"$ .

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Consider $\"\"$ .

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

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$\"\"$

\

$\"\"$ .

\

Consider $\"\"$ .

\

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

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$\"\"$

\

$\"\"$ .

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

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$\"\"$

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$\"\"$ .

\

$\"\"$

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

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$\"\"$

\

$\"\"$

\

$\"\"$

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

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If $\"\"$ then $\"\"$ .

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If $\"\"$ then $\"\"$ .

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$\"\"$

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$\"\"$

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Substitute $\"\"$ .

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$\"\"$

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$\"\"$

\

$\"\"$

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Apply the 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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The surface area of the soild obtained after rotation of the minor axis is

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$\"\"$ .

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$\"\"$

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(a)The surface area of the soild obtained after rotation of the major axis is

\

$\"\"$ .

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(b) The surface area of the soild obtained after rotation of the minor axis is

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$\"\"$ .