Simple Region Question for a Double Integral Substitution

[AFinch]

Homework Statement

Evaluate the double integral integral ∫∫2x^2-xy-y^2 dxdy for the region R in the first quadrant bounded by the lines y=-2x+4, y=-2x+7, y=x-2, and y=x+1 using the transformation x=1/3(u+v), y=1/3(-2u+v).

Homework Equations

The Attempt at a Solution

I've obtained the Jacobian (it's 1/3) and I've plugged in the transformation equations into the line equations to get 4<=v<=7 and -1<=u<=2.

My question is pretty simple: if the region R is in the first quadrant, does the transformed region also need to be restricted to the first quadrant? As I'm typing this and thinking about it, It doesn't really make sense to restrict the transformed region to the first quadrant, but if someone could confirm that I would appreciate it.

 

[LCKurtz]

If the xy region is R, the integral over R transforms in uv space to the integral over whatever region R transforms to with the substitution.