Integrating to find the volume of a finite region

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SUMMARY

The discussion focuses on calculating the volume of the finite region enclosed by the surfaces z = 0 and x² + y² + z = 1 using triple integration. Participants emphasize the importance of visualizing the region through sketching and suggest that due to the symmetry of the problem, a single integration may suffice. The key steps involve determining the appropriate limits for x, y, and z based on the defined surfaces.

PREREQUISITES
  • Understanding of triple integration in calculus
  • Familiarity with the concept of volume under surfaces
  • Knowledge of cylindrical coordinates for integration
  • Ability to sketch 3D surfaces and regions
NEXT STEPS
  • Learn about setting up triple integrals for volume calculations
  • Study the use of cylindrical coordinates in integration
  • Research techniques for visualizing 3D regions in calculus
  • Explore examples of symmetric volume calculations
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Students and educators in calculus, mathematicians focusing on integration techniques, and anyone interested in geometric volume calculations.

james525
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Find the volume of the finite region enclosed by the surfaces z = 0 and
x2 + y2 + z = 1

I know I have to do triple integration on dV to accomplish this but do not know where to start and what limits to use for x, y and z?

Cheers guys
 
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welcome to pf!

hi james! welcome to pf! :smile:

(try using the X2 icon just above the Reply box :wink:)
james525 said:
I know I have to do triple integration on dV to accomplish this but do not know where to start and what limits to use for x, y and z?

no, it's symmetric, so you can get away with a single integration! :wink:

first sketch the region (so that you know what it looks like), then slice it into very thin slices whose area you already know :smile:
 

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