Change of variables to evaluate the integral

[yis]

Homework Statement

these are on the picture

Homework Equations

transformation, jacobian

The Attempt at a Solution

I don't know how to enter the equation, so i uploaded the picture..
is it alright?
I think I solved it somehow, but not quite sure if it is right...
please tell me if there's a wrong process.
thank you..

if you can't see the image, this link would be ok..
http://image.kilho.net/?pk=1599118

 

[UltrafastPED]

The link to the image reports back with "401 Access Denied."

 

[haruspex]

Looks ok to me (and the link works).

 

[dauto]

Looks right except for the limits of the integrals at the very last line that seem wrong.

 

[HalfLostAndSearching]

I understand the importance of using appropriate change of variables techniques to evaluate integrals in order to simplify complex calculations. In this problem, the transformation and Jacobian methods can be used to change the variables in the given integral and make it easier to solve.

Firstly, we can use the transformation method to change the variables by setting u = x + y and v = x - y. This transformation allows us to rewrite the given integral as an integral over u and v, which can be easier to integrate. However, we must also take into account the change in the limits of integration for u and v.

Next, we can use the Jacobian method to determine the relationship between the original variables (x and y) and the new variables (u and v). This involves calculating the determinant of the Jacobian matrix, which represents the partial derivatives of u and v with respect to x and y. This will help us to properly adjust the limits of integration for the new variables.

By using these techniques, we can simplify the given integral and solve it with ease. However, it is important to carefully check our work and make sure all steps are correct in order to obtain an accurate solution.