Undergrad Finding new region for double integral

[Mr Davis 97]

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]

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

 

[Mr Davis 97]

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]

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.

 

[Mr Davis 97]

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]

What are x and y?

 

[Mr Davis 97]

What are x and y?

Sorry, replace x with u and y with v.

 

[mfb]

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