$\"\"$

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

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Moments and center of mass of a planar lamina:

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Let $\"\"$ and $\"\"$ be continuous functions such that $\"\"$ on $\"\"$ , and consider the planar lamina of uniform density $\"\"$ bounded by the graphs of $\"\"$ and $\"\"$ .

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The moments about the $\"\"$ -axis and $\"\"$ -axis are

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$\"\"$ .

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$\"\"$ .

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The center of mass $\"\"$ is $\"\"$ and $\"\"$ , where $\"\"$ is the mass of the lamina.

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$\"\"$ \ \

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Graph: \ \

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

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$\"\"$

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Observe the graph:

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The curves intersects on $\"\"$ .

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

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$\"\"$ on $\"\"$ .

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$\"\"$

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Moment about $\"\"$ - axis is $\"\"$ .

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

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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Moment about $\"\"$ - axis is $\"\"$ .

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

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$\"\"$

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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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Center of mass $\"\"$ .

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$\"\"$

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