$\"\"$ \ \

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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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Determine the points of intersection by plotting their graphs.

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

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Graph:

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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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Find $\"\"$ .

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

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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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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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Find $\"\"$ .

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

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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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$\"\"$

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$\"\"$ . \ \

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$\"\"$ \ \

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Find the center of mass $\"\"$ .

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

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

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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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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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$\"\"$ .

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The center of mass $\"\"$ .

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$\"\"$

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$\"\"$ , $\"\"$ and $\"\"$ . \ \