Multiple integration + Centroid Help.

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Homework Help Overview

The discussion revolves around finding the volume bounded by a sphere and a paraboloid, as well as locating the centroid of the resulting region. The subject area includes multiple integration techniques and geometric interpretations in three-dimensional space.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to set up an integral for the volume but expresses difficulty with the integration process, particularly with the variable r. Some participants suggest using Fubini's theorem to evaluate the integral step-by-step, while others inquire about the original poster's specific calculations to provide feedback.

Discussion Status

The discussion is ongoing, with participants exploring different methods of integration and questioning the original poster's results. There is a request for clarification on the centroid calculation, indicating that multiple aspects of the problem are being examined without a clear consensus on the approach.

Contextual Notes

The original poster mentions confusion regarding evaluating integrals and expresses a need for guidance on both the volume and centroid calculations. This indicates a potential lack of familiarity with the integration techniques required for the problem.

numberonenacho
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Homework Statement


Find the volume bounded by sphere rho = rt. 6 and the paraboloid z = x^2 + y^2
and locate the centroid of this region


The attempt at a solution

http://www.mathhelpforum.com/math-help/latex2/img/4deb41286077aabd94b30802f0e6a68a-1.gif

So Thats the integral that I made for this problem, but I'm having trouble integrating it. the rdr throws it off.

Please Help~! I am having some trouble.
 
Last edited by a moderator:
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You first integrate with z, that gives you rz in the inside. Using Fubini you get (meaning evaluate the integral) [tex]r\sqrt{6-r^2} - r(r^2)[/tex] and now continue with r and theta.
 
So when i evaluate it, I get -12pi rt6 as the answer.
Am I doing it wrong? I feel like I am not doing right. If you could, could I see how someone would do a problem like this? Evaluating integrals is a little confusing for me.

And then I need lots of help on the centroid part as well. Thanks~
 
Why don't you show us what you did and we can comment on it.
 

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