# Volume between sphere and outside cylinder.

haruspex
Homework Helper
Gold Member
2020 Award
I don't think cylindrical coordinates look at all bad. x^2+y^2=2x has a pretty simple form in cylindrical coordinates.
Sure, but the ranges of integration get nasty when you get into the part where the sphere cuts through the ends of the cylinder.

Dick
Homework Helper
In any case, I believe (0,0) corresponds to theta=pi/2, (+-1/2,+-sqrt(3)/2) corresponds to theta=pi/3. Is that correct? Isn't the range of theta [0,pi/2] U [1.5*pi,2pi]?

Yeah. That range looks good. And ok at pi/3 if you take the + sign. The - sign point isn't on the circle.

Last edited:
Dick
Homework Helper
Sure, but the ranges of integration get nasty when you get into the part where the sphere cuts through the ends of the cylinder.

Seems to work out ok for me. The domain of the intersection of the cylinder with the xy plane is inside that of the sphere.

haruspex
Homework Helper
Gold Member
2020 Award
Seems to work out ok for me. The domain of the intersection of the cylinder with the xy plane is inside that of the sphere.
No, I mean where the sphere cuts through the cylinder, defining its ends.

Dick
Homework Helper
No, I mean where the sphere cuts through the cylinder, defining its ends.

As peripetain had (sort of) in the first post, the z values of the ends are where z=+/-sqrt(4-r^2). That's good enough, isnt it?

Last edited:
haruspex
Homework Helper
Gold Member
2020 Award
In cylindrical, you have z, r, theta. What will be your integration order?
As a check on the answer, numerically I get about 9.66 for the volume removed by the cylinder, leaving 23.85 in the sphere. Sound right? Still working on the analytic result using Cartesian.

Dick
Homework Helper
In cylindrical, you have z, r, theta. What will be your integration order?
As a check on the answer, numerically I get about 9.66 for the volume removed by the cylinder, leaving 23.85 in the sphere. Sound right? Still working on the analytic result using Cartesian.

I get 9.644049708034451 for the volume removed. So that sounds about right. I integrated dz first, dr second and dtheta third. peripatein had the right general idea in the first post, except that the r factor in the measure was left out, the z limits are wrong, the r limits are wrong and the theta limits are also wrong. It just needs to be fixed. But seriously, if you are still working on the result in Cartesian, I'd give it up unless it's become an obsession. It's HARD to do that way. It's not THAT much of a challenge in cylindrical.

Last edited:
haruspex