Change of Variable for Double Integral: Jacobian's Integration [SOLVED]

[Suitengu]

[SOLVED] Jacobian's Integration

Homework Statement

Find an appropriate change of variable in order to evaluate the double integral over R. (Hint:you could find the equations of the boundary lines, and then do the change of variables)

$$\int R \int 4(x+y)\exp{x-y} dA$$

Homework Equations

The Attempt at a Solution

I don't understand how I am going to change the variable when I received no region to begin with. I am wondering if this was a mistake on the part of the teacher.

 

[HallsofIvy]

This problem makes no sense unless you are given "R". Perhaps there was a picture to go with the problem?

 

[Suitengu]

There was no picture whatsoever and I like you was dumb-struck. I thought it was just a question which was clearly out of my league but hearing this response let's me know she must have made a mistake. *sighs* ah brotha

 

[Suitengu]

I looked through some text and I found the problem but this one has the equations
x = (1/2)(u+v) and y = (1/2)(u-v). But she said we should solve the problem as an indefinite integral so i am now lost.

 

[HallsofIvy]

I really would like to know what the problem really was! From x= (1/2)(u+ v) and y= (1/2)(u- v) we can get u+ v= 2x and u- v= 2y so, adding the equations, 2u= 2x+ 2y and u= x+ y. Subtracting the two equations, 2v= 2x- 2y and v= x- y. I strongly suspect that the original problem was to integrate over a diamond shaped region with boundaries given by x+ y= constant and x- y= constant.

 

[Suitengu]

yeah she showed us in class today that the problem originally stemmed with a triangular region. But she had expected us to bring some equations out of thin air like u = x+y and v = u-v and find the jacobian and do an indefinite integral.

In case of the triangular area with vertices: (-1 1) (0 0) and (1 1), how would you do a change of variable to make that look like a square or rectangular region in terms of u an v?