Use double integrals to calculate the area inside the ellipse whose semiaxes have length a and b

The general equation of an ellipse centered at the origin with its semiaxes coinciding with the coordinate axes is given by



Substituting and into the above equation gives the Pythagorean identity, so we can use polar coordinates quite nicely to our advantage.

If is the ellipse with the equation above, the area is given by the double integral



Let and , so that the Jacobian matrix is



and the magnitude of its determinant is

since in polar coordinates we use the convention that , and because they are lengths.

Now, the area is given by