Special integration of Sphere Volume

[TommyC]

Firstly, I'm a newby to PF. Secondly, I've studied calculus (2 semesters) but it's been >35 years.

I'm interested in deriving a simple formula for computing the partial volume of a sphere. Such a volume would be from the surface to some point along its radius toward the center. Crudely, suppose you could whack off (along a plane) a piece of a sphere, invert it and fill it w/ liquid. If you knew the piece's depth, how much liquid would it hold?

From Wiki (http://en.wikipedia.org/wiki/Sphere), I've captured the integration to the attached a MS Word file (I'm not sure how to capture the piece w/ its equation to this comment field and have it look right). Maybe someone can reply and bring it into this comment field.

I believe the sense of the integration implied by the formula is from the center of the sphere to some point (x) along its radius toward the surface. My interest is just the reverse.

In any case, if someone could please help me out w/ a simple formula in terms of the radius and depth of the severed piece, that would be fantastic. (I can well imagine this was probably on one of my tests in integral calculus but I can no longer remember. )

 

[Ben Niehoff]

I don't open unfamiliar .doc files. However, you can find the answer to your question here:

http://mathworld.wolfram.com/SphericalCap.html

And with that, you should be able to work backwards and fix up your integration.

 

[TommyC]

Perfect, Ben. The link you provided refers to the shape as a spherical cap (unsophisticated, I was thinking of a soup bowl).

Not sure how it's derived from integration (academic to my main interest) but Eqn. 2 is exactly what I'm looking for.

As a curiosity, it note in Eqn. 1 that 'b' goes to zero, eliminating the middle term. What's 'b'? In the figure, it's not shown.

 

[TommyC]

My earlier curiosity notwithstanding ('b' in Eqn. 1), if any of you are feeling really lazy, here's a super simple way to compute the volume of the spherical cap:

http://www.1728.com/sphere.htm

 

[wolf1728]

My spherical cap calculator has a new URL - it is now http://www.1728.org/sphere.htm
The reason? My domain name 1728.com was stolen and I don't know when or even if I'll regain it. I figured people at this forum should know about that. It seems you folks like to link to my site quite a bit - and thanks.