(This was a question from my calculus final this semester.)

Find the volume bounded by the surface

and the first octant.

Using the change of variables

the Jacobian of the transformation is

.

So now, I tried to find volume by calculating the triple integral over the constraints

.

What I'm not sure about is the bounds of the regions; here's what I have:

For the original region E,

, and for the transformed region T(E),

.

Is that right? I'm confused by the positive/negative signs that result from squaring and square roots.