# sphere-sphere intersection

Let

$S_1 : (x-1)^2 +y^2+z^2=1$

$S_2 : x^2 +y^2 +z^2 =1$

$S_3 : (x+1)^2 +y^2 +z^2 =1$

Find the volume of the solid inside $S_2$ and outside $S_1$ and $S_3$, using triple integrals.

I have try to express the region enclosed by two spheres in spherical coordinates, but I am stuck... Could anyone help me please ?

• I don't think spherical coordinates will be useful here, since the problem has no spherical symmetry. – celtschk Apr 11 '15 at 14:16
• This is straightforward if you use one-variable calculus to set up a volume of rotation about the $x$-axis. (It may be easier to find the volume of intersection of two of the balls, then subtract appropriately. A sketch will certainly help.) – Andrew D. Hwang Apr 11 '15 at 15:11
• $$\verb/Volume/ = 2 \int_0^{\frac12} \left[\pi(1-x^2) - \pi(1-(1-x)^2)\right]dx = \cdots$$ – achille hui Apr 11 '15 at 15:46