If you already know some multivariable calculus, you can simply compute the double integral
where denotes the region bounded by the ellipse with equation
We can solve for :
then the integral becomes
We also could have solve for instead and swapped the order of integration, so that the area is
If you don't know about double integrals yet, these basically reduce to either of the single-variate integrals,
(making use of the fact that is symmetric about 0) or
either of which can be evaluated with a trigonometric substitution. For instance, taking , gives , and the integral becomes
The integral with respect to can be resolved in a similar way.
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We also could have converted to polar coordinates first, parameterizing the region by
The Jacobian matrix for this transformation is
and its determinant gives . So the integral reduces to