Finding Volumes of Sphere & Circular Cone: Alpha from 0 to Pi

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Discussion Overview

The discussion revolves around finding an expression for the volume enclosed between a sphere of radius 1 centered at the origin and a circular cone with a half-angle alpha, also centered at the origin. Participants explore the mathematical formulation of this volume integral and consider specific limits for alpha.

Discussion Character

  • Homework-related, Mathematical reasoning

Main Points Raised

  • One participant suggests using a volume integral to derive the expression for the volume between the sphere and the cone.
  • Another participant speculates that when alpha = pi, the volume will be half of the sphere's volume.
  • A participant shares a link to a Wikipedia page on spherical sectors, which may provide relevant information.
  • There is a reminder that homework or homework-like problems should be posted in the appropriate sections, indicating a concern about the thread's placement.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the approach to solving the problem, and multiple viewpoints regarding the limits of alpha and the expected volume values remain present.

Contextual Notes

Some assumptions regarding the definitions of the volume integral and the geometric configurations may be implicit. The discussion does not resolve the mathematical steps necessary to derive the volume expression.

MaryCatherine
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Use an appropriate volume integral to find an expression for the volume enclosed between a sphere of radius 1 centered on the origin and a circular cone of half-angle alpha with its vertex at the origin. Show that in the limits where alpha = 0 and alpha = pi that your expression gives the expected values.

Any ideas or clues? Where alpha = pi I think the volume will be half the volume of the sphere.
 
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that's actually very helpful, thanks
 
MaryCatherine said:
Use an appropriate volume integral to find an expression for the volume enclosed between a sphere of radius 1 centered on the origin and a circular cone of half-angle alpha with its vertex at the origin. Show that in the limits where alpha = 0 and alpha = pi that your expression gives the expected values.

Any ideas or clues? Where alpha = pi I think the volume will be half the volume of the sphere.
Please post homework or homework-like problems in the Homework & Coursework sections, not here in the technical math sections.
 

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