Cartesian to Polar in Double Integral

[manenbu]

Homework Statement

Solve:
$$\iint_{\frac{x^2}{a^2}+\frac{y^2}{b^2} \leq 1} \sqrt{1-\frac{x^2}{a^2}-\frac{y^2}{b^2}} dx dy$$

Homework Equations

Cartesian to Polar

The Attempt at a Solution

Well - this Integral should be solved as a polar function (the radical should be $$\sqrt{1-r^2}ab$$ when expressed in polar coordinates. I just can't get this right. Guidance please?

 

[HallsofIvy]

Homework Statement

Solve:
$$\iint_{\frac{x^2}{a^2}+\frac{y^2}{b^2} \leq 1} \sqrt{1-\frac{x^2}{a^2}-\frac{y^2}{b^2}} dx dy$$

Homework Equations

Cartesian to Polar

The Attempt at a Solution

Well - this Integral should be solved as a polar function (the radical should be $$\sqrt{1-r^2}ab$$ when expressed in polar coordinates. I just can't get this right. Guidance please?

No, the radical is NOT of that form in polar coordinates. But you can modify the coordinate system to "elliptic coordinates". Let $x= r a cos(\theta)$, $y= r a sin(\theta)$. Then $x^2/a^2= r^2 cos^2(\theta)$ and $y^2/b^2= r^2 sin^2(\theta)$. You will need to use the Jacobian to get the correct differential dx dy is not just "$r dr d\theta$" now.

 

[manenbu]

ok, so maybe the "ab" is part of the differential. I guess that it comes from the Jacobian, which I have no idea how to use.