Calculating volume between two paraboloids

[Physgeek64]

Homework Statement

A volume is enclosed by the plane z = 0 and the inverted paraboloid, z = 6 − r2 (expressed in cylindrical coordinates). Find the volume and its surface area.

Hence, using a suitable linear transformation, find the volume of the region enclosed between the surfaces z = ax^2 +by^2 and z = 6−cx^2 −dy^2 where a, b, c and d are positive constants

Homework Equations

below

The Attempt at a Solution

So I've managed to do the first part (Photo of my working will be below), but am now struggling to calculate the volume between the two paraboloids. I thought of one possible parameterisation in which u^2=b/a x^2 and v^2=d/c x^2 so that I transform the two elliptical paraboloids into regular ones, but this would mean I have a 4-dimentional coordinate system, which is obviously not correct. Its mainly the parameterisation I'm struggling with

Many thanks in advance :)

 

[Physgeek64]

My working- Part 2

 

[Physgeek64]

My working- Part 1

 

[Physgeek64]

I was thinking- maybe I could subtract them and let u=sqrt(a+c)x, v= sqrt(b+d) y ? I couldn't make it work- but is that along the right lines? Many thanks