Volume of a sphere without cap

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SUMMARY

The discussion focuses on calculating the volume of a solid sphere with a spherical cap removed from its north pole. The volume of the sphere is established as \( \frac{4}{3} \pi R^3 \). The volume of the spherical cap is derived using the formula \( \frac{\pi h^2}{3}(3R - h) \), where \( h \) is the height of the cap. Participants emphasize the importance of understanding the apex angle \( \alpha \) and suggest using integration techniques to derive the volume of the cap directly.

PREREQUISITES
  • Understanding of solid geometry, specifically spheres and spherical caps.
  • Familiarity with calculus, particularly integration techniques.
  • Knowledge of the formula for the volume of a sphere: \( \frac{4}{3} \pi R^3 \).
  • Ability to interpret geometric parameters such as apex angles in conical shapes.
NEXT STEPS
  • Learn how to derive the volume of a spherical cap using integration techniques.
  • Study the relationship between apex angles and the geometry of cones.
  • Explore advanced solid geometry concepts, including the derivation of volumes for various shapes.
  • Investigate practical applications of spherical volume calculations in physics and engineering.
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Students studying geometry, mathematics educators, and anyone interested in advanced calculus applications related to solid shapes and volumes.

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Homework Statement


A solid sphere of radius R has a spherical cap, defined by the cone theta = alpha, removed from its "north pole". Determine the volume of the sphere without cap.


Homework Equations





The Attempt at a Solution


Well obviously, the volume would be volume of sphere - volume of cap.
I am able to derive the volume of the sphere and obtain the formula 4(pi)r^3/3, but I am not sure about the volume of the cap. Is this the same cap that is talked about? http://en.wikipedia.org/wiki/Spherical_cap
If so, the volume would (pi)h^2/3(3r-h), and the answer could be found by subtracting the two quantities. My class focuses on deriving the quantities though, and I am confused and want to understand on where to start for deriving the volume of the cap.
 
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what does "theta=alpha" mean?
Do you mean that the apex angle of the cone is ##\alpha##? The half-angle at the apex?

The cap is the regeon described in the wikipedia entry you linked to - yes.
Why not use the same method for the volume of the cap that you used for the volume of the sphere?

For that matter - why not do it in one go by carefully choosing your limits of integration?
 

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