Thread: Nearly impossible integration problem!

How do I find the volume common to two spheres, each with radius r, if the center of each sphere lies on the surface of the other sphere?

Originally Posted by MarkOfEternity

How do I find the volume common to two spheres, each with radius r, if the center of each sphere lies on the surface of the other sphere?

This doesn't make sense. It would make more sense if you mean to say:

How do I find the volume common to two spheres, each with radius r, if the center of one sphere lies on the surface of the other sphere?

No, because if they have the same radius, if the centre of one is on the surface of the other, the centre of the other will be on the surface of the first.

Agreed. Since the radius of each sphere is similar, then the outside of each sphere must touch the center point of the other. In any case, I've been trying to pound out an answer. I found something online that says I should try , but I can't figure out how they came to that conclusion or really even what to do with it.

You can calculate the volume for part of a sphere using discs right? For example, The plot below is a cross-section of two spheres with radius 2 with the properties you required. Suppose you wanted to calculate the volume of the blue part using the disc method. Wouldn't that be:

.

Now, can you figure the x-coordinate when the two spheres intersect for arbitrary r, and using the same method, come up with an expression for the total volume common to both spheres with radius r?

I think this is what I needed to see to get this problem done. I appreciate the reply. Thanks for the help.