$\"\"$

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

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(a)

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Find the area of the region.

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Find the integral limits by equating two curve equations.

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$\"\"$

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$\"\"$

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$\"\"$ .

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Area of the region bounded by $\"\"$ , $\"\"$ , $\"\"$ and $\"\"$ .

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

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The two curves $\"\"$ on interval $\"\"$ .

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The area enclosed by the curves is $\"\"$ .

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$\"\"$ .

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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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$\"\"$

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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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(b)

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Find the volume of the solid generated by revolving the region about the $\"\"$ -axis.

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The volume of the solid generated revolving about the $\"\"$ - axis.

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Formula for the volume of the solid with the Washer method,

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$\"\"$ .

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The outer radius of revolution is $\"\"$ .

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The inner radius of revolution is $\"\"$ .

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$\"\"$

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$\"\"$

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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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$\"\"$

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Again apply integration by parts.

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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 volume of the solid generated by revolving the region about the $\"\"$ -axis is $\"\"$ .

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$\"\"$

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(c)

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Find the volume of the solid generated by revolving the region about the $\"\"$ -axis.

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The volume of the solid generated revolving about the $\"\"$ -axis is $\"\"$ .

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

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$\"\"$ .

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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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$\"\"$

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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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The volume of the solid generated by revolving the region about the $\"\"$ -axis is $\"\"$ .

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$\"\"$

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(d)

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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 $\"\"$ -axis are $\"\"$ . $\"\"$ .

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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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Substitute $\"\"$ .

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$\"\"$ .

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$\"\"$ .

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

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$\"\"$

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Substitute $\"\"$ .

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$\"\"$ .

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$\"\"$ .

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

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$\"\"$

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(a) The area of the region is $\"\"$ .

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(b) The volume of the solid generated by revolving the region about the $\"\"$ -axis is $\"\"$ .

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(c) The volume of the solid generated by revolving the region about the $\"\"$ -axis is $\"\"$ .

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(d) The centroid of the region is $\"\"$ .