$\"\"$

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The equations is $\"\"$ , $\"\"$ , $\"\"$ and $\"\"$ .

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(a) Find the volume of the region about the $\"\"$ -axis.

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Disk method:

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The volume of the solid is $\"\"$ .

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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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The equations is $\"\"$ , $\"\"$ , $\"\"$ and $\"\"$ .

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(b) Find the volume of the region about the $\"\"$ -axis.

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Shell method:

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Vertical axis of revolution.

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The volume of the solid is $\"\"$ .

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The distance from the center of the rectangle to the axis of revolution is $\"\"$ .

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The height of the rectangle is $\"\"$ .

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

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The graph is symmetrical about the $\"\"$ -axis.

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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 equations are $\"\"$ , $\"\"$ , $\"\"$ and $\"\"$ .

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(c) Find the volume of the region about the line $\"\"$ .

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Shell method:

\

The volume of the solid is $\"\"$ .

\

The distance from the center of the rectangle to the axis of revolution is $\"\"$ .

\

The height of the rectangle is $\"\"$ .

\

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

\

The graph is symmetrical about the $\"\"$ -axis.

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$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

$\"\"$

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(a) The volume of the solid about the $\"\"$ -axis is $\"\"$ .

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

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(c) The volume of the solid about the line $\"\"$ is $\"\"$ .