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?

Printable View

- Oct 13th 2009, 08:33 PMMarkOfEternityNearly 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?

- Oct 13th 2009, 09:59 PMChris L T521
- Oct 13th 2009, 10:05 PMMatt Westwood
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. - Oct 13th 2009, 11:02 PMMarkOfEternityAdditional Details
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 $\displaystyle dv = 2 * [pi] x^2 dy$, but I can't figure out how they came to that conclusion or really even what to do with it.

- Oct 14th 2009, 07:58 AMshawsend
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:

$\displaystyle V=\int_1^2 \pi r^2 dx=\int_1^2 \pi (\sqrt{4-x^2})^2 dx=\int_1^2 \pi (4-x^2)dx$.

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? - Oct 14th 2009, 08:22 AMMarkOfEternityThanks
I think this is what I needed to see to get this problem done. I appreciate the reply. Thanks for the help.