$\"\"$

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

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Find the centroid of the region.

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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 $\"\"$ -axes are $\"\"$ . $\"\"$ .

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

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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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Consider $\"\"$ .

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Solve the integral using integration by parts.

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Formula for integration by parts: $\"\"$ .

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Here $\"\"$ and $\"\"$ .

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Consider $\"\"$ .

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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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Consider $\"\"$ .

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Apply integral on each side.

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$\"\"$

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$\"\"$ .

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Substitute corresponding values in $\"\"$ .

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$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

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

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$\"\"$

\

$\"\"$

\

$\"\"$ .

\

$\"\"$

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Find $\"\"$ .

\

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

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

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Consider $\"\"$ .

\

Solve the integral using integration by parts.

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Formula for integration by parts: $\"\"$ .

\

Here $\"\"$ and $\"\"$ .

\

Consider $\"\"$ .

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

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$\"\"$

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$\"\"$

\

$\"\"$ .

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Consider $\"\"$ .

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Apply integral on each side.

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$\"\"$

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$\"\"$ .

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Substitute corresponding values in $\"\"$ .

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$\"\"$

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$\"\"$

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Again apply integration by parts.

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$\"\"$

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$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

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

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

$\"\"$

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Find $\"\"$ .

\

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

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

Consider $\"\"$ .

\

Solve the integral using integration by parts.

\

Formula for integration by parts: $\"\"$ .

\

Here $\"\"$ and $\"\"$ .

\

Consider $\"\"$ .

\

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

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$\"\"$

\

$\"\"$

\

$\"\"$ .

\

Consider $\"\"$ .

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Apply integral on each side.

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$\"\"$

\

$\"\"$ .

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Substitute corresponding values in $\"\"$ .

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$\"\"$ .

\

Again apply integration by parts.

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Here $\"\"$ and $\"\"$ .

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

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$\"\"$

\

$\"\"$

\

Again apply integration by parts.

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Consider $\"\"$ .

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Here $\"\"$ and $\"\"$ .

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

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

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Substitute $\"\"$ in

\

$\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

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$\"\"$

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

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The centroid is $\"\"$ .

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Observe the example 6 :

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The function $\"\"$ is the inverse function of the $\"\"$ and the region is same.

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The centroid of example 6 is $\"\"$ .

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Therefore, the centroid is also inverse.

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$\"\"$

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The centroid is $\"\"$ .