Calculating volume between two paraboloids

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Homework Help Overview

The problem involves calculating the volume enclosed between two paraboloids, specifically the inverted paraboloid z = 6 - r² and another paraboloid defined by the equation z = ax² + by². The context is set in cylindrical coordinates, and the task includes finding a suitable linear transformation to facilitate the calculation of the volume between these surfaces.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • The original poster attempts to parameterize the problem but expresses confusion regarding the dimensionality of their approach. They suggest a transformation involving u² = b/a x² and v² = d/c x², questioning the validity of this method. Another participant proposes a different transformation involving subtraction of the two paraboloids and suggests letting u = sqrt(a+c)x and v = sqrt(b+d)y, but is uncertain about its effectiveness.

Discussion Status

Contextual Notes

Participants are navigating the complexities of transforming the equations of the paraboloids and are grappling with the implications of their parameterization choices. The discussion reflects a need for clarity on the dimensional aspects of the transformations being considered.

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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-dimensional coordinate system, which is obviously not correct. Its mainly the parameterisation I'm struggling with

Many thanks in advance :)
 
Last edited:
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My working- Part 2
 

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My working- Part 1
 

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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
 

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