Volume of a sphere with a hole through it

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Homework Help Overview

The discussion revolves around calculating the volume of a sphere with a cylindrical hole through its center. The sphere has a radius of 2, while the hole has a radius of 1. Participants are exploring various methods to approach the problem, including geometric interpretations and integration techniques.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Participants are considering different methods to find the volume, such as revolving shapes, using polar caps, and employing shell or washer methods. There are questions about the appropriate equations for the semicircle and the integration process for the polar caps.

Discussion Status

The discussion is active, with participants providing hints and suggestions for different approaches. Some participants have noted potential issues with certain methods, while others are sharing diagrams to aid understanding. There is no explicit consensus on a single approach, but multiple lines of reasoning are being explored.

Contextual Notes

Participants mention constraints related to the integration of polar caps and the challenges of setting up the equations correctly. There is also a reference to homework rules that may limit the type of guidance provided.

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

http://www.berrys.plus.com/hh2.gif
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! :wink:
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 by a moderator:
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.
 
BerryBoy said:
I'm assuming you haven't completed this question yet...
Correct me if I'm wrong.
http://www.berrys.plus.com/hh2.gif
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! :wink:
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

Well I had forgotten all about this problem. I had already figured out the height 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.
 
Last edited by a moderator:

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