Double integral change of variables

[V0ODO0CH1LD]

Homework Statement

Use the change of variables $u=x+y$ and $y=uv$ to solve:
$$\int_0^1\int_0^{1-x}e^{\frac{y}{x+y}}dydx$$

Homework Equations

The Attempt at a Solution

So I got as far as:
$$\int\int{}ue^vdvdu.$$

But I just can't find the region of integration in terms of $u$ and $v$.

 

[CAF123]

What does the domain of integration look like in x-y space? Then consider what happens to points on the interior and boundary of that domain under the transformation to u-v space.

 

[V0ODO0CH1LD]

the region looks like a triangle, basically the lower half of the unit square. But how do I transform this region into uv space? I tried to solve the equations of change of variables until I either got the bounds of $u$ as two numbers and the bounds of $v$ at least in terms of $u$. Or the other way around.. I figured I could solve the integral either way.

 

[CAF123]

the region looks like a triangle, basically the lower half of the unit square. But how do I transform this region into uv space? I tried to solve the equations of change of variables until I either got the bounds of $u$ as two numbers and the bounds of $v$ at least in terms of $u$. Or the other way around.. I figured I could solve the integral either way.

Yes, you can solve u=u(x,y) and v=v(x,y). Then choose points on the boundary of the domain in xy space and see what the corresponding point is in uv space. A point in the interior of the domain in xy space will point you towards the interior of the region in uv space.