$\"\"$

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(a)

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The curves are $\"\"$ and $\"\"$

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$\"\"$

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Substitute $\"\"$ in the above equation

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$\"\"$

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The direction of the closed loop is towards origin.

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The limits of $\"\"$ is $\"\"$

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Find area:

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Area is region enclosed by the astroid equai to $\"\"$ into area of the region in the first quadrant.

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Area is region enclosed the $\"\"$ - axis is $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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Area is enclosed by the curve is $\"\"$ .

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$\"\"$

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(b)

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Find volume:

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The curves are $\"\"$ and $\"\"$

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Consider, $\"\"$

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

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$\"\"$

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$\"\"$

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$\"\"$

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Change the limits

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$\"\"$

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The volume of the solid is above $\"\"$ - axis is $\"\"$ .

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Substitute $\"\"$ and $\"\"$ in the formula.

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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

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$\"\"$

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(c)

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Find the centroid.

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By symmetry, the $\"\"$ -coordinate of the centroid is zero.

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$\"\"$

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Replace $\"\"$ .

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$\"\"$

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Substitute $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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Substitute $\"\"$ in the above equation

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$\"\"$

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The centriod is $\"\"$ .

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$\"\"$

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(a)Area is enclosed by the curve $\"\"$ is $\"\"$

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(b) The volume of the solid is $\"\"$ is $\"\"$

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(c) The centriod is $\"\"$ .