Calculus 3 Triple Integration in Spherical Coords

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Homework Help Overview

The discussion revolves around evaluating a triple integral in spherical coordinates, specifically the integral of z over a solid defined between two spheres, x²+y²+z²=1 and x²+y²+z²=4.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to set up the integral with specific boundaries for spherical coordinates but expresses uncertainty about the conceptual meaning of the result, particularly when the integral yields zero.

Discussion Status

Some participants have provided insights regarding the nature of the integral, suggesting that the positive and negative contributions may cancel out. There is acknowledgment of the correct boundaries for the radial coordinate, but no consensus has been reached on the implications of the result.

Contextual Notes

The original poster indicates a lack of clarity regarding the conceptual understanding of triple integrals that do not represent volume, which may affect their interpretation of the results.

Wargy
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Homework Statement



Use spherical coordinates to evaluate the triple integral z dV where Q is the solid that lies between x^2+y^2+z^2=1 and x^2+y^2+z^2=4.

Homework Equations


Not sure what goes here :P

The Attempt at a Solution


I've gotten everything set up, I am having problems with boundaries I think. Currently I am using 0 to 2[tex]\pi[/tex] for [tex]\vartheta[/tex], 0 to [tex]\pi[/tex] for [tex]\varphi[/tex] and 1 to 2 for [tex]\rho[/tex]. When solving, I get zero as my final answer, and since I'm not clear on the conceptual meaning of a triple integral that isn't of a function that equals 1 (volume) I don't know if this answer makes sense.
 
Last edited:
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It is zero. It's the integral of z on the volume between two spheres centered on the origin. It's as much positive as negative. The two cancel. But you should be using rho from 1 to 2.
 
Nevermind, read what you wrote again.
Thanks for the quick reply!
 
Last edited:
Makes a lot of sense.
 

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