Calculating Volume of Sphere Intersection Using Cavalieri's Principle

[DavidSnider]

This is not homework, just a toy problem I was thinking about.

Let's say you have two spheres, one larger than the other. The center of the smaller sphere is placed on the surface of the larger sphere. How would you find the volume of the intersection?

 

[Dr. Lady]

I can't think of any way to do this pre-calculus, but I can guarantee that you could do it with calculus. Take circular slices of the solid going along the radius pointing toward the center of the large sphere, find an equation for the Area of the circles based on how far you are along the radius (needs to be piecewise for which sphere you're in) and then integrate the area function over length.

 

[Defennder]

There was a similar problem concerning area of intersection of circles which was solved by the ancient Greeks, who certainly didn't have access to calculus. Can't remember what it's called or whether the Greeks managed to extend their method to cover the 3D version of a sphere.

 

[mathwonk]

archimedes knew how to find the volume of a segment of a sphere and your problem is the sum of two such segments. his methods were essentially integral calculus, but without the antidifferentiation of the fundamental theorem.

i.e. he had what we now call cavalieri's principle and used that.