[QuantumOverlord]

Okay, two spheres are intersecting, they have different radii. One sphere is NOT inside the other, how do I use volume integration in spherical coordinates to find the volume of intersection.

Thankyou so much

[nuodai]

Take the line joining their two centres; call this your 'x-axis'.

Now the boundaries of the two spheres intersect in a circle (except in the trivial case where the volume of the region of intersection is zero), and your 'x-axis' passes through the centre of the circle; choose any diameter of this circle, and call it your 'y-axis'. Choose the z-axis appropriately.

Now take the cross-section in the (x,y)-plane. Notice that this is symmetric about the z-axis, and that the cross-section consists of two circles that intersect on the y-axis. The volume of the region of intersection will be what you get when you revolve the relevant region lying above the x-axis by a full turn around the x-axis.

[QuantumOverlord]

(Original post by nuodai)
Take the line joining their two centres; call this your 'x-axis'.

Now the boundaries of the two spheres intersect in a circle (except in the trivial case where the volume of the region of intersection is zero), and your 'x-axis' passes through the centre of the circle; choose any diameter of this circle, and call it your 'y-axis'. Choose the z-axis appropriately.

Now take the cross-section in the (x,y)-plane. Notice that this is symmetric about the z-axis, and that the cross-section consists of two circles that intersect on the y-axis. The volume of the region of intersection will be what you get when you revolve the relevant region lying above the x-axis by a full turn around the x-axis.

Thanks :P

I found the plane of intersection then integrated using the appropiate values of r, for my sphere cap, I did this on my other sphere cap by recentering the origin at the centre of the 2nd sphere. I found the range of the azimuthal angle then rotated around the x axis, finally I summed the volume of these two caps.

Is that right?

Thx