The discussion focuses on converting a double integral from Cartesian to polar coordinates for the equation \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. The correct approach involves using elliptic coordinates, specifically defining x and y as x = r a cos(θ) and y = r b sin(θ). Participants emphasize the importance of applying the Jacobian to adjust the differential, indicating that the standard polar form does not apply directly in this case.
PREREQUISITESStudents and educators in mathematics, particularly those studying calculus and integral transformations, as well as anyone involved in advanced mathematical modeling requiring coordinate system changes.
No, the radical is NOT of that form in polar coordinates. But you can modify the coordinate system to "elliptic coordinates". Let [itex]x= r a cos(\theta)[/itex], [itex]y= r a sin(\theta)[/itex]. Then [itex]x^2/a^2= r^2 cos^2(\theta)[/itex] and [itex]y^2/b^2= r^2 sin^2(\theta)[/itex]. You will need to use the Jacobian to get the correct differential dx dy is not just "[itex]r dr d\theta[/itex]" now.manenbu said:Homework Statement
Solve:
[tex]\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[/tex]
Homework Equations
Cartesian to Polar
The Attempt at a Solution
Well - this Integral should be solved as a polar function (the radical should be [tex]\sqrt{1-r^2}ab[/tex] when expressed in polar coordinates. I just can't get this right. Guidance please?