Evaluate the Flux with Divergence Theorem

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Homework Help Overview

The discussion revolves around evaluating the flux of a vector field using the Divergence Theorem, specifically for the vector field F = <(e^z^2, 2y + sin(x^2z), 4z + (x^2 + 9y^2)^(1/2)> within a defined region bounded by the inequalities x^2 + y^2 < z < 8 - x^2 - y^2.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants discuss the application of the Divergence Theorem and the computation of the divergence of the vector field. There are concerns about the limits of integration for the variables r and z, with some participants questioning their correctness.

Discussion Status

Some participants express uncertainty regarding the limits of integration and seek verification of their setup. There is an acknowledgment of potential errors, and one participant suggests a revised bound for r, indicating an ongoing exploration of the problem.

Contextual Notes

Participants are working under the constraints of the problem's geometric setup, specifically the volume defined by the intersection of two paraboloids, which may influence their calculations and assumptions.

PsychonautQQ
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Homework Statement


Evaluate the flux where F = <(e^z^2,2y+sin(x^2z),4z+(x^2+9y^2)^(1/2)> in the boundary of the region x^2 + y^2 < z < 8-x^2-y^2


Homework Equations





The Attempt at a Solution


So using the divergence Theorem,

∇ dot F = 6

∫∫∫6r dzdrdθ

where z is bounded between r^2 and 8-r^2
and r is bounded between 0 and 8^(1/2)
and θ is bounded between 0 and 2∏

is this correct?
I'm mainly worried about my limits of integration for r and z are incorrect, can anyone verify?
 
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Okay so yeah what I wrote is probably wrong, can anyone help me and my friends along?
 
BRAIN BLAST!
I just realized that the r bound should go from 0 to 2? right??
 
PsychonautQQ said:
BRAIN BLAST!
I just realized that the r bound should go from 0 to 2? right??
The enclosed volume is the volume between two paraboloids. I think what you have is correct.
 

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