$\"\"$

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The graphs of the inequalities are $\"\"$ , $\"\"$ , $\"\"$ and $\"\"$ .

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

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$\"\"$

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Observe the graph:

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

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$\"\"$

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Moments and center of mass of a planar lamina:

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

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

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

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Find the area of the region $\"\"$ .

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$\"\"$

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$\"\"$ .

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Substitute $\"\"$ .

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Apply derivative on each side with respect to $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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Substitute $\"\"$ 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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$\"\"$

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$\"\"$

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Substitute $\"\"$ 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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$\"\"$

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$\"\"$ .

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The area of the region is $\"\"$ .

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$\"\"$

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Find $\"\"$ .

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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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Since the graph is symmetrical about the $\"\"$ -axis, center of mass lies on the axis of symmetry $\"\"$ .

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$\"\"$

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The center of mass $\"\"$ is $\"\"$ and $\"\"$ .

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

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$\"\"$ .

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

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$\"\"$ .

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The centroid of the region is $\"\"$ .

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$\"\"$

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The centroid of the region is $\"\"$ .