Parabolic coordinate system question

[Felicity]

Homework Statement

a 3D solid is bounded by 2 paraboloids. The binding condition in cartesian coordinates is

-1+(x2+y2) < 2z < 1-(x2+y2)

a) rewrite the binding condition in parabolic coordinates
b) using parabolic coordinates and the (already derived) metric tensor, find the volume of the solid

Homework Equations

x=stcos(p) y= stsin(p) z= (t2-s2)/2

The Attempt at a Solution

I found the binding conditions to be equal to

-1 + s2t2 < t2 - s2 < 1 - s2t2

I have the metric tensor and I know i just need to do a triple integral and multiply by the square root of the metric tensor but how do I find the functions of s, t and p and how do I know the limits of integration?

I've tried splitting it into two inequalities and moving the variables around looking for a pattern but I can't really see anything.


any help would be greatly appreciated

thank you

-Felicity

 

[Felicity]

Never mind