$\"\"$

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The equations of the curves are $\"\"$ and $\"\"$ and $\"\"$ .

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Determine the points of intersection by plotting their graphs.

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Graph:

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Draw the coordinate plane.

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Graph the curves $\"\"$ and $\"\"$ .

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$\"\"$

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Observe the graph:

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The two curves intersect at the points $\"\"$ and $\"\"$ .

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$\"\"$

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Consider $\"\"$ and $\"\"$ .

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Moment about $\"\"$ - axis $\"\"$ of a planar lamina: $\"\"$ .

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Where $\"\"$ uniform density of a planar lamina.

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Here $\"\"$ value varies form $\"\"$ to $\"\"$ , so $\"\"$ and $\"\"$ .

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Substitute $\"\"$ , $\"\"$ and limits of integral in the above formula.

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$\"\"$

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$\"\"$ .

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$\"\"$

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Moment about $\"\"$ - axis $\"\"$ of a planar lamina: $\"\"$ .

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Substitute corresponding values in the formula.

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$\"\"$

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$\"\"$ .

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$\"\"$

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Center of mass $\"\"$ and $\"\"$ , where $\"\"$ .

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Mass of the system is $\"\"$ .

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

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$\"\"$

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$\"\"$ .

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Substitute $\"\"$ , $\"\"$ and $\"\"$ values in center of mass formula.

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$\"\"$

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$\"\"$ .

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$\"\"$ .

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Center of mass $\"\"$ .

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$\"\"$

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Moments $\"\"$ , $\"\"$ and center of mass $\"\"$ .