Integrating Norm in Unit Ball in Rn-2

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    Integrating Norm
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SUMMARY

The discussion centers on integrating the function \(\int |x|^2\) with respect to the vector \(x\) in the unit ball in \(\mathbb{R}^{n-2}\). The user initially applied a change of variable to the last two components and utilized Fubini's Theorem but encountered difficulties in performing the integration. They inquired about the use of generalized spherical coordinates, specifically questioning if \(|x|^2\) equates to \(r^2\). Ultimately, the user found an alternative solution, indicating a resolution to their query.

PREREQUISITES
  • Understanding of Fubini's Theorem
  • Familiarity with integration in \(\mathbb{R}^n\)
  • Knowledge of spherical coordinates in multiple dimensions
  • Basic concepts of vector calculus
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  • Research the application of Fubini's Theorem in higher dimensions
  • Study generalized spherical coordinates in \(\mathbb{R}^n\)
  • Explore techniques for integrating functions over unit balls
  • Learn about alternative methods for solving integrals in vector calculus
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Mathematicians, students studying multivariable calculus, and anyone interested in advanced integration techniques in higher-dimensional spaces.

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\int|x|2 with respect to the vector x in the unit ball in Rn-2

I'm dealing with volumes of unit balls in Rn and after applying a change of variable to the last 2 components and Fubini's Theorem, I get that integral and can't find a way to integrate it. Any help on this?
 
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Generalized spherical coordinates.
 
Would that just make |x|2=r2? I've never worked with generalized spherical coordinates before, so could you elaborate a bit?
 
Nevermind, I found an alternate way.
 

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