# Double integral change of variables

## Homework Statement

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

## 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
Gold Member
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.

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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
Gold Member
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.