Parametrized surfaces

[It401301]

Let
S be the surface defined by z= 1/ (x2 + y2)^(1/2)
for

z ≥ 1.
(a) Sketch the graph of this surface.
(b) Show that the volume of the region bounded by
S
and the plane
z = 1 is finite. (You will need to use
an improper integral.)
(c) Show that the surface area of
S is infinite.

I understand that the surface looks like a volcano. I assume x=s, y=t, and z=f(s,t), but I don't know exactly what the limits of integration are for t and s other than the fact that the hint that it goes from 1 to infinity. Its obviously of the form dsdt and follows from the fact that I have to take the line integral of the magnitude of the normal vector Ts x Tt. At least, that would be my assumption for how to solve c. I don't even know how to begin finding the volume. Many thanks in advance for your help.

 

[MaxJasper]

Take a look at Torricelli Tower: volume infinite integration converges, but surface area integration diverges!

 

[It401301]

Take a look at Torricelli Tower: volume infinite integration converges, but surface area integration diverges!

I think I sort of see it. z=1/r. It is a series of concentric circles like Gabriel's horn. Double integral where theta is 0 to 2 pi r goes from 1 to 1/r of rdrdtheta.

Although I don't think that gives me volume.

It could be x=s, y=t and z=1/(s2+t2)^(1/2)
and then I could find Tt and Ts. But I don't know what
my bounds would be in that scenario.