# Finding new region for double integral

I have a double integral where the region of integration is ##R = \{(u,v) : 0 \le u < \infty, ~0 \le v < \infty) \}##. I am doing the change of variables ##u=zt## and ##v = z(1-t)##. I am a bit rusty on calc III material, so how would I find the new region of integration, in terms of the zt-coordinate system?

mfb
Mentor
Did you draw a sketch?
Did you express u=0 and v=0 in the new coordinates?

Did you draw a sketch?
Did you express u=0 and v=0 in the new coordinates?
Will I know that in uv coordinates the region is the first quadrant.

If u=0, then z=v.
If v=0, then z=u.

Does this help me in any way?

mfb
Mentor
If u=0, then z=v.
If v=0, then z=u.
These are not the only options, and you are trying to do two steps at the same time. Do it step by step.

These are not the only options, and you are trying to do two steps at the same time. Do it step by step.
If ##x=0,y \ne 0##, then ##z=y,t=0##
If ##y=0,x \ne 0##, then ##z=x, t=1##.

Is this correct?

mfb
Mentor
What are x and y?

What are x and y?
Sorry, replace x with u and y with v.

mfb
Mentor
And if u=v=0 then z=0.
That looks good and it should tell you what to integrate over.