Word Count : 425

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We are to find the volume of the ellipsoid using double integration, given: $$ \frac{x^2}{4}+\frac{y^2}{9}+\frac{z^2}{16}=1 $$ --- ### Step 1: Express $z$ in terms of $x$ and $y$ Rearranging: $$ \frac{z^2}{16} = 1 - \frac{x^2}{4} - \frac{y^2}{9} $$ $$ z^2 = 16 \left( 1 - \frac{x^2}{4} - \frac{y^2}{9} \right) $$ $$ z = \pm 4 \sqrt{ 1 - \frac{x^2}{4} - \frac{y^2}{9} } $$ The volume will be: $$ V = \iint_{R} (z_\text{top} - z_\text{bottom}) \, dx\,dy $$ $$ V = \iint_{R} \left[ 4 \sqrt{ 1 - \frac{x^2}{4} - \frac{y^2}{9} } - (-4 \sqrt{ 1 - \frac{x^2}{4} - \frac{y^2}{9} }) \right] dx\,dy $$ $$ V = \iint_{R} ____ ____ ______ _____ ____ __________ _____ _______.
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