(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.