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

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SUMMARY

The discussion focuses on evaluating the triple integral $\int\int\int_H(x^2+y^2) dV$, where the region H is defined by the boundaries x² + y² = 1, y = x, y = 0, z = 0, and z = 2. Participants recommend using cylindrical coordinates for simplification, noting that the conversion leads to the integral involving r². The importance of the Jacobian in the transformation process is emphasized, ensuring accurate evaluation of the integral.

PREREQUISITES
  • Cylindrical coordinates in multivariable calculus
  • Understanding of triple integrals
  • Knowledge of Jacobian determinants
  • Familiarity with the concept of moment of inertia
NEXT STEPS
  • Study the application of cylindrical coordinates in triple integrals
  • Learn about Jacobian transformations in multivariable calculus
  • Explore the calculation of moments of inertia for various shapes
  • Practice evaluating triple integrals with different boundary conditions
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Students and professionals in mathematics, particularly those studying calculus and integral evaluation, as well as engineers and physicists interested in applications of triple integrals in real-world scenarios.

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How would I evaluate the triple integral \int\int\int_H(x^2+y^2) 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 cos^{2}(x) + sin^{2}(x) = 1 and you should be left with r^{2}. Making the integration easier. Don't for get the Jacobian.
 
Cheers. Thought so
 

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