Definite integrals calculus. can someone me on where to get started with this. prove the area of this ellipse

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

step-by-step explanation:

when the line is solid:

when the line is dashed:

this > inequality sign means:

this < inequality sign means:

this > inequality sign means:

this < inequality sign means:

about the problem:

srry i'm not done ! i'll

the correct answer would be c