Volume of a sphere with a hole through it

1. Nov 9, 2005

Swatch

I have to calculate the volume of a sphere of radius 2 that has a hole with radius 1 through the sphere and that includes the center of the sphere. I am trying to solve this by putting semi disk with lenght 4 units and 1 unit from the base to the top, and then revolving this disk around the line y=1. Then I will get a sphere of radius 2 with a hole through it with radius 1. I don't know if this is a good approach but I have a problem with the equation of the semi circle. I don´t know how to equate this semicircle, since y=sqrt(a^2-x^2) is not working. Could someone please give me a hint?

2. Nov 9, 2005

Tide

I would try to find the volume of the "polar caps" that you effectively cut off say at the north and south poles to form your cylindrical hole. Then subtract that and the volume of the fully enclosed cylinder from the volume of the starting sphere to get the volume of what remains.

3. Nov 9, 2005

BerryBoy

I'm assuming you haven't completed this question yet...
Correct me if I'm wrong.

I agree with Tide - as you know the volumes for the other shapes. Consider this diagram and integrate to find the "polar caps". Unfortunately your idea of integrating round y=1 would not work but your way of thinking is good!
Now I've drawn the diagram (which I believe every physics/maths solution should have where applicable), let me know how you get on...

Regards,
Sam

Last edited: Nov 10, 2005
4. Nov 9, 2005

NateTG

It seems to me that 'shells' or 'washers' would also be a good way to do this problem, since the holy sphere is a solid of rotation.

5. Nov 12, 2005

Swatch

Well I had forgotten all about this problem. I had already figured out the hight of the polar caps. But I have a problem with integrating the caps to find their volumes, could you please give me a hint on that? Berry Boy.
Thanks.