Spherical Coordinates: Integrating a Hemisphere/Paraboloid

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Homework Help Overview

The discussion revolves around converting a given integral setup from rectangular coordinates to spherical coordinates, specifically focusing on a hemisphere and a paraboloid. The original poster describes the integration limits and expresses uncertainty about how to proceed with the conversion and visualization in spherical coordinates.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • The original poster attempts to visualize the problem in both 2D and 3D, questioning how to represent the shapes involved in spherical coordinates. They express confusion about the next steps and whether to consider a 'slice' of the shape. Other participants inquire about the specific question being addressed and suggest exploring the relationship between the variables in the context of the paraboloid.

Discussion Status

The discussion is ongoing, with some participants providing guidance on the relationship between the variables in spherical coordinates, particularly in relation to the paraboloid's surfaces. There is no explicit consensus yet, as the original poster continues to seek clarity on the conversion process.

Contextual Notes

The original poster has not fully articulated the specific question or task associated with the integral setup, which may affect the direction of the discussion. There is an acknowledgment of the complexity introduced by the nature of the paraboloid in spherical coordinates.

Tater
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Homework Statement
The outermost integral is:
-2 to 2, dx

The middle integral is:
-sqrt(4-x^2) to sqrt(4-x^2), dy

The inner most integral is:
x^+y^2 to 4, dz






The attempt at a solution

Drawing the dydx in a simple 2d (xy) plane, it is circular with a radius of 2. So this means that the period(theta) will go from 0 to 2pi. Drawing in 3d (xyz) yields a hemisphere/paraboloid. Now this is where I'm stuck. I don't know what to do after this or how to really tackle this problem. Do I want to attempt to draw a 'slice' of it in the spherical outline with the variables phi, rho, theta? Do I have to look at it a certain way (2d or 3d)? I just don't see what I can do!

Any help or guidance is greatly appreciated!
 
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You haven't actually said what the question asks you to do..
 
Whoops. Thought I stuck that in there. Anyways, all I have to do is convert it to spherical coordinates (from rectangular to spherical).
 
Tater said:
Whoops. Thought I stuck that in there. Anyways, all I have to do is convert it to spherical coordinates (from rectangular to spherical).

I was afraid of that :wink:

It is indeed a paraboloid, so [itex]\rho[/itex] and [itex]\phi[/itex] will not be independent the way they would if it was a spherical section...

Try finding the relationship between [itex]\rho[/itex] and [itex]\phi[/itex] for the paraboloid's curved and flat surfaces
 
From my weblog
http://buyanik.wordpress.com/2009/05/02/laplacian-in-spherical-coordinates/"
 
Last edited by a moderator:

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