How to Solve Integrals with Change of Variables and Jacobians?

[robierob]

I need to find out how to solve this integral with the indicated changes/transformations.

int.[0 to 2/3]int[y to 2-2y]
(x+2y)e^(y-x)
dxdy

u=x+2y v=x-y

I know that the xy region is x=y y=0 and y=1- (x/2)
which is a triangle

so I created systems with U and V but can't get a new bounded region...

I just get u=v and two other lines that intersect at the origin.
If anyone can tell me what's up with this it would be helpfull.

Also, does it seem weird that my given v doesn't match the problem exactly?
the u and v were given changes.

 

[HallsofIvy]

I need to find out how to solve this integral with the indicated changes/transformations.

int.[0 to 2/3]int[y to 2-2y]
(x+2y)e^(y-x)
dxdy

u=x+2y v=x-y

I know that the xy region is x=y y=0 and y=1- (x/2)
which is a triangle

so I created systems with U and V but can't get a new bounded region...

I just get u=v and two other lines that intersect at the origin.
If anyone can tell me what's up with this it would be helpfull.

Also, does it seem weird that my given v doesn't match the problem exactly?
the u and v were given changes.

What does the region look like in the xy-coordinate system?

One line is y= x. If v= x- y, what is v on that line? Another line is y= 1- (1/2)x or 2y= 2- x so x+ 2y= 2. What is u on that line? Finally, the third line is y= 0. In that case, u= x and v= x so y= 0 corresponds to the line u= v. What does that look like in the uv-coordinate system?

 

[robierob]

Thanks

Thanks, that put me back on track!