Volume of a spherical segment

  • #1
I have a problem in a math book that says "Use Cavalieri's Principle to find the volume of a spherical segment of one base and thickness h if the radius of the sphere is r."

I believe it looks like this:
http://img395.imageshack.us/img395/2826/sphere1.jpg [Broken]

how do i solve it?
 
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Answers and Replies

  • #2
HallsofIvy
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"Cavalieri's principle" says "If, in two solids of equal altitude, the sections made by planes parallel to and at the same distance from their respective bases are always equal, then the volumes of the two solids are equal ".

In order to use that you would have to find some other solid, whose volume is easy to get, having the same altitude, h, and sections of the same area. I don't see any simple way to do that.

You could, of course, find the area by dividing the segment into disks of thickness dx. The radius of each disk can be found by looking at the right triangle formed by a radius of the disk, a radius of the sphere, and the vertical line through the centers of the disks.
 
  • #3
the best i can do so far is to think of a cone with base radius (r-h) and height (h), where the inverse of such cone should be the volume of the spherical segment. but that is incorrect.

however, i am pretty sure that this is the right track.
 

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