Converting triple integral to spherical

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Homework Help Overview

The discussion revolves around converting a triple integral into spherical coordinates, focusing on the limits of integration for a specific integral related to a sphere of radius 1.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the integration region and the limits for spherical coordinates, questioning the interpretation of the shape as a hemisphere and the corresponding limits for rho, phi, and theta.

Discussion Status

Participants are actively exploring the limits of integration and the shape of the region of interest. Some guidance has been offered regarding the interpretation of the spherical coordinates, but there is still uncertainty about the conventions used for the angles.

Contextual Notes

There is a focus on the integration region being limited to the positive x-axis, and participants are considering the implications of this on the limits for phi and theta in spherical coordinates.

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Well the best way of checking the limits in a case like this is to draw the regions you are integrating over (ignoring the integrand).

What is the integration region of the first integral?

I'll help you with this one: x goes from 0 to 1, z goes from -sqrt(1-x^2) to sqrt(1+x^2) so these two integrals trace out a semicircle in the x-z plane (in the x>0 region). Since y goes from -sqrt(1-x^2-z^2) to sqrt(1-x^2-z^2) this rotates the semicircle along the y-axis to give a...


What is the second (transformed) integration region?

Are these the same? If so good, if not change the limits so they are.
 
I don't really get what you are saying.

I know that sqrt(1-x^2-z^2) and -sqrt(1-x^2-z^2) makes a sphere with radius 1, which can be turned into rho=1.

I just can't figure out what phi and theta run to. Since I think it's a sphere (right?) can I assume phi and theta run complete cycles? Is that right?

Can someone please tell me these two limits (and if rho does indeed =1) so that I can see if I'm doing this right?
 
Wait, if it's a sphere of rho=1, it then projects a shadow on the x-z plane of a circle with radius one, but the only area of interest is from x=0 to x=1.

So a semi-sphere that exists only in the positive-x?

Is that the shape and the correspondingly correct limits?
 
hemi-sphere in positive x sounds good to me, now what do your spherical limits become...
 
rho from 0 to 1

phi from 0 to pi

and theta from 0 to pi

Right?
 
Right!?
 
depends on your convention if phi is the angle around the horizontal & the x-axis aligns with phi = pi/2, which i think it does, then yes
 

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