How could you possibly solve the equation without knowing what
u,v,w are? If you know what they are then you might have a hope of solving it. So by saying "solve
z2=x2+y2 for
u,v,w", you've said as much as you can until you know what
u,v,w are. But the question is asking a general question, so you're not expected to give an explicit form of an answer (not least because such an expression doesn't exist); the specifics come later on.
For instance, we could use cylindrical polar coordinates
(u,v,w)=(r,φ,z), so that we have
x=rcosφ,
y=rsinφ and
z=z. Then the equation of the surface,
z2=x2+y2, becomes
z2=r2. Note that this doesn't depend on
φ, so
φ can take any value and as long as
r=∣z∣ the equation is still satisfied.
(This makes sense since all the equation above tells you is that for a given z, the cross-section of the graph through that point on the z-axis is just a circle of radius |z|, which is why it looks like a cone [or rather two cones stuck together at their tips].) So we can integrate over
θ,z by taking
1≤z≤2 (which is the given limit) and
0≤θ<2π, and setting
r=z in the integral and multiplying by the Jacobian.
This is just one example of a parametrization. (It just happens to be probably the easiest parametrization to use in this case.)