$\"\"$

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(a)

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The polar curve is $\"\"$ , $\"\"$ .

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Formula10.2.6: Surface area is given by $\"\"$ .

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As the polar curve is the function of $\"\"$ .

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Replace $\"\"$ by $\"\"$ in the formula.

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$\"\"$ .

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Polar to rectangular coordinates relation: $\"\"$ and $\"\"$ .

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Consider $\"\"$ .

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Differentiate with respect to $\"\"$ on each side.

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$\"\"$

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Consider $\"\"$ .

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Differentiate with respect to $\"\"$ on each side.

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$\"\"$ .

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Here $\"\"$ .

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Substitute corresponding valuees in expression (1).

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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(b)

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The polar curve is $\"\"$ .

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$\"\"$

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Differentiate with respect to $\"\"$ on each side.

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$\"\"$

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Polar curve is rotating about $\"\"$ axis.

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So consider only top part area of the curve then by the symmetry double the result.

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$\"\"$ .

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Substitute corresponding valuees in expression.

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$\"\"$

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$\"\"$

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$\"\"$ sq.units.

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$\"\"$

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(a) $\"\"$

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(b) $\"\"$ sq.units.