Evaluating Triple Integral with Change of Variable: Help

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Discussion Overview

The discussion focuses on evaluating the triple integral \(\int\int\int_G x+y+z \, dV\) using a change of variables, specifically within the region defined by the inequalities \(0 \leq x+y \leq 1\), \(2 \leq y+z \leq 3\), and \(4 \leq x+z \leq 5\). Participants explore the transformation of variables and the implications for the integral's evaluation.

Discussion Character

  • Mathematical reasoning
  • Technical explanation
  • Homework-related

Main Points Raised

  • One participant proposes using the change of variables \(u = x+y\), \(v = y+z\), and \(w = x+z\) to simplify the integral.
  • Another participant suggests that after determining the Jacobian, the next step is to express \(x+y+z\) in terms of \(u\), \(v\), and \(w\).
  • A hint is provided regarding the relationship \(u+v+w = 2(x+y+z)\), leading to the expression \(x+y+z = (u+v+w)/2\).
  • Participants express enthusiasm about deriving the relationship between the variables.

Areas of Agreement / Disagreement

Participants generally agree on the approach of using a change of variables and the derived relationship between \(x+y+z\) and the new variables \(u\), \(v\), and \(w\). However, the discussion does not resolve the complete evaluation of the integral.

Contextual Notes

There are no explicit limitations noted, but the discussion does not cover all steps necessary for the complete evaluation of the integral, such as the final integration limits after the change of variables.

squenshl
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How do I evaluate the triple integral [tex]\int\int\int_G[/tex] x+y+z dV using a suitable change of variable where G is the region
0 [tex]\leq[/tex] x+y [tex]\leq[/tex] 1, 2 [tex]\leq[/tex] y+z [tex]\leq[/tex] 3, 4 [tex]\leq[/tex] x+z [tex]\leq[/tex] 5.
I know to let u = x+y, v = y+z, w = x+z and I end up with the
det(jac) = |2| [tex]\Rightarrow[/tex] 1/det(jac) = |1/2|. But I'm stuck after that. Help.
 
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Hi squenshl! :wink:
squenshl said:
How do I evaluate the triple integral [tex]\int\int\int_G[/tex] x+y+z dV using a suitable change of variable where G is the region.

Well, you've got the bounds, and you know how to rewrite the dV (from the Jacobian), so all you need is to rewrite x+y+z in terms of u v and w, which is … ? :smile:
 
Hint:

What does u+v+w equal, in terms of x+y+z?
 
u+v+w = 2x+2y+2z = 2(x+y+z),
[tex]\Rightarrow[/tex] x+y+z = (u+v+w)/2.
Then just chuck that in. Is that right. Thanks.
 
Last edited:
squenshl said:
x+y+z = (u+v+w)/2.

:biggrin: Woohoo! :biggrin:​
 
Cheers.
 

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