$\"\"$

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The equation of the ellipse is $\"\"$ .

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(a)

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Find the area of the region bounded by the ellipse.

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$\"\"$ .

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Rewrite the equation.

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$\"\"$

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$\"\"$ represents the one-half part of the ellipse.

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Find the limits of integration.

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$\"\"$

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Integrate between $\"\"$ to $\"\"$ .

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Area of the ellipse is $\"\"$ .

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$\"\"$

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$\"\"$

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Integral formula: $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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(b)

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Disk method:

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

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In this case $\"\"$ .

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The integral limits are $\"\"$ and $\"\"$ .

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$\"\"$

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$\"\"$

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Power rule of integration: $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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Find the surface area of the ellipse.

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Surface area with respect to major axis: $\"\"$ .

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$\"\"$ .

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Apply derivative with respect to $\"\"$ .

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$\"\"$

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Integrate between $\"\"$ and $\"\"$ .

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$\"\"$

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$\"\"$

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Integral formula: $\"\"$ .

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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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(c)

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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 integral limits are $\"\"$ and $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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Power rule of integration: $\"\"$ .

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$\"\"$

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$\"\"$ .

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$\"\"$

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Find the surface area of the ellipse.

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Surface area with respect to minor axis: $\"\"$ .

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$\"\"$ .

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Rewrite the equation.

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$\"\"$

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Apply derivative with respect to $\"\"$ .

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$\"\"$

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Equate $\"\"$ to zero.

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$\"\"$

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Integrate between $\"\"$ and $\"\"$ .

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$\"\"$

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Integral formula: $\"\"$ .

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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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(a) $\"\"$

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(b)

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$\"\"$ .

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$\"\"$

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(c)

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$\"\"$ .

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$\"\"$