$\"\"$

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The base of a solid $\"\"$ is an elliptical region $\"\"$ .

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Cross sections perpendicular to the $\"\"$ -axis are isosceles right triangles with hypotenuse on base.

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Draw the top view of the solid with parallel cross sections of length as $\"\"$ and radius of the base $\"\"$ is on -axis..

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$\"\"$

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Observe the figure,

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From the pythagotrean theorem, $\"\"$ .

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Consider the side of the isosceles triangle as $\"\"$ .

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$\"\"$

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$\"\"$ .

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Area of the cross section is $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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Consider $\"\"$ .

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$\"\"$

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$\"\"$ .

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

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$\"\"$ .

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$\"\"$

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Find the volume of the solid by integrating the area with respect to $\"\"$ over the limits $\"\"$ to $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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Volume of the solid is $\"\"$ .

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$\"\"$

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Volume of the solid is $\"\"$ .