Evaluating Triple Integral: $\int\int\int_H(x^2+y^2) dV$

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Discussion Overview

The discussion revolves around evaluating the triple integral \(\int\int\int_H(x^2+y^2) dV\), where \(H\) is defined as the region bounded by \(x^2 + y^2 = 1\), \(y = x\), \(y = 0\), \(z = 0\), and \(z = 2\). The focus is on the mathematical approach to solving this integral, particularly through the use of cylindrical coordinates.

Discussion Character

  • Mathematical reasoning

Main Points Raised

  • One participant suggests using cylindrical coordinates to evaluate the integral, indicating that it simplifies the process.
  • Another participant notes that the integral relates to the Moment of Inertia about the z-axis and emphasizes the importance of the identity \(\cos^{2}(x) + \sin^{2}(x) = 1\) in the conversion to cylindrical coordinates, leading to \(r^{2}\) in the integral.
  • A reminder is given about the necessity of including the Jacobian when changing to cylindrical coordinates.

Areas of Agreement / Disagreement

Participants generally agree on the approach of using cylindrical coordinates, but there is no consensus on the specific evaluation steps or the interpretation of the integral's physical meaning.

Contextual Notes

Some assumptions regarding the region \(H\) and the limits of integration may not be fully detailed, and the discussion does not resolve the exact evaluation of the integral.

squenshl
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How would I evaluate the triple integral [tex]\int\int\int_H(x^2+y^2)[/tex] dV,
where H is the region bounded x2 + y2 = 1, y = x, y = 0, z = 0, z = 2
 
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It is sector of a cylinder. Try cylindrical coordinates. A very simple integral results.
 
I was about to say Moment of Inertia about the z axis. But no!.

Um yes when you convert to cylindrical co-ordinates remember the [tex]cos^{2}(x) + sin^{2}(x) = 1[/tex] and you should be left with [tex]r^{2}[/tex]. Making the integration easier. Don't for get the Jacobian.
 
Cheers. Thought so
 

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