$\"\"$

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

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$\"\"$ .

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The general form of ellipse is $\"\"$ .

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Comapre the equation with $\"\"$ .

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Here, $\"\"$ and $\"\"$ .

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The parametric representation of the general ellipse is $\"\"$ and $\"\"$ , $\"\"$

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Susbtitute $\"\"$ and $\"\"$ .

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

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$\"\"$

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Arc length :

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The arc length of the curve for the parametric equations $\"\"$ and $\"\"$ in the interval $\"\"$ is $\"\"$ .

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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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Consider $\"\"$ .

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Apply derivative on each side with respect to $\"\"$ .

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$\"\"$ .

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Susbtittute $\"\"$ and $\"\"$ and $\"\"$ in the function $\"\"$ .

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$\"\"$

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$\"\"$

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Substittute $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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The Simpsons Rule for approximating $\"\"$ is given by

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$\"\"$ ,

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where $\"\"$ and $\"\"$ .

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Fidn the $\"\"$ for $\"\"$ .

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Since we are integrating the half part of the integral, hence the value of $\"\"$ .

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Susbttitute $\"\"$ and $\"\"$ .

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$\"\"$

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Susbtitute $\"\"$ .

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$\"\"$

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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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$\"\"$ .

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$\"\"$ .

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$\"\"$

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Substittue all the values in $\"\"$ .

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$\"\"$

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$\"\"$

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The length of the circumference of the ellipse is $\"\"$ .

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$\"\"$

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The length of the circumference of the ellipse is $\"\"$ .