Solving Jacobean Problem: Integrating (x+y) over x and y variables

[mrkb80]

Homework Statement

I'm also having trouble with this Jacobean problem. I really could use some help:

Evaluate ∫∫(x+y)dxdy over y=x,y=x-5,y=-x,x+y=5

Homework Equations

The Attempt at a Solution

I know that if I can get u and v correct this becomes a simple integral, but I have no idea what to make u and v and how to set my limits

 

[Dick]

Take a wild guess. Express your limit conditions by moving all of the x's and y's to one side and the constants to the other. What sort of expressions do you see on the x and y side?

 

[HallsofIvy]

The boundary of the region are parallel straight lines. In fact, they are at right angles so this is a rectangle. You want to change it to a rectangle with sides along the coordinate axes. So define u and v so that the equations of the sides become u= constant and v= constant. That is what Dick is suggesting you do.

 

[mrkb80]

so if I am understanding correctly, y-x=5 and x+y=5 therefore setting u=x+y and v=y-x so that u=5 and v=5, but how do i prove that the other sides of the rectangle are x=0 and y=0 (or u=0 and v=0)