Evaluating Double Integrals Using Elliptical Rescaled Polar Coordinates

When dealing with double integrals over certain regions, particularly those with elliptical or radial symmetry, employing elliptical rescaled polar coordinates can greatly simplify the calculation process. In this article, we will explore the method of transforming a general region into a simpler form using these coordinates, and evaluate a particular double integral as an application.

Transformation to Elliptical Rescaled Polar Coordinates

Consider the transformation from Cartesian coordinates (x, y) to elliptical rescaled polar coordinates (r, theta) using the following formulas:

[x a r cos theta, quad y b r sin theta]

Here, r leq 1, and theta in [0, frac{pi}{2}]. This transformation facilitates the integration over an elliptical region. To perform the transformation properly, we need to calculate the Jacobian of this coordinate system.

Jacobian of the Transformation

The Jacobian of the transformation is given by:

[frac{partial(x, y)}{partial(r, theta)} begin{vmatrix} a cos theta -ar sin theta b sin theta br cos theta end{vmatrix} abr]

Using this Jacobian, we can rewrite the double integral iint_R x^2 y^2 , dx , dy as follows:

[iint_R x^2 y^2 , dx , dy int_0^{pi/2} int_0^1 a r cos theta^2 cdot b r sin theta^2 cdot ab r , dr , dtheta]

Step-by-Step Evaluation of the Double Integral

Let's break this down step-by-step:

[iint_R x^2 y^2 , dx , dy int_0^{pi/2} int_0^1 a^3 r^3 cos theta^2 cdot b^2 r^2 sin theta^2 , dr , dtheta] [ a^3 b^3 int_0^{pi/2} int_0^1 r^5 cdot sin theta cos theta^2 , dr , dtheta] Integrating with respect to r: [ a^3 b^3 int_0^{pi/2} int_0^1 r^5 , dr , dtheta cdot sin theta cos theta^2] [ a^3 b^3 int_0^{pi/2} left[frac{1}{6} r^6 right]_0^1 , dtheta cdot sin theta cos theta^2] [ frac{1}{6} a^3 b^3 int_0^{pi/2} sin theta cos theta^2 , dtheta] Using the identity for simplification: [sin theta cos theta^2 frac{1}{4} sin 2theta (1 - cos 4theta)] Substituting back into the integral: [ frac{1}{24} a^3 b^3 int_0^{pi/2} sin^2 2theta , dtheta] Integration by parts (or a table of integrals can be used): [int_0^{pi/2} sin^2 2theta , dtheta frac{pi}{4}] Substituting the result back: [ frac{1}{24} a^3 b^3 cdot frac{pi}{4} frac{pi}{96} a^3 b^3]

Conclusion

Thus, the value of the double integral iint_R x^2 y^2 , dx , dy is given by:

[boxed{frac{pi}{96} a^3 b^3}]

Evaluating double integrals using elliptical rescaled polar coordinates can be a powerful technique for simplifying complex integrations. This method not only simplifies the integrand but also the domain of integration, making the process more straightforward and efficient.

Keywords: elliptical coordinates, polar coordinates, double integrals