$\"\"$

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Observe the figure :

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Vertex of the ellipse are $\"\"$ .

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The ellipse is symmentric about the $\"\"$ axis.

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Therefore the area is equal to $\"\"$ times the area of the top portion, which is given by the area of the top portion is integral of the function from $\"\"$ to $\"\"$ .

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$\"\"$

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Top portion area is also is symmentric about the $\"\"$ axis.

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Therefore the area is equal to $\"\"$ times the area of the first quadrant from $\"\"$ to $\"\"$ .

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$\"\"$

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$\"\"$

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Apply the trignometric substitution to evaluate the integral.

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Let $\"\"$ .

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$\"\"$

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When $\"\"$ , $\"\"$ then $\"\"$ .

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When $\"\"$ , $\"\"$ then $\"\"$ .

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Therefore the limits are $\"\"$ to $\"\"$ .

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$\"\"$

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$\"\"$ .

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$\"\"$

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Consider $\"\"$ .

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Evaluate the function by splitting the integral into two parts.

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Apply the formula : $\"\"$ .

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$\"\"$

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$\"\"$

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

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$\"\"$

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