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

[MaryCatherine]

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.

 

[fresh_42]

https://en.wikipedia.org/wiki/Spherical_sector

 

[MaryCatherine]

that's actually very helpful, thanks

 

[Mark44]

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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