Volume of a sphere with a hole through it

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SUMMARY

The discussion focuses on calculating the volume of a sphere with a radius of 2 that has a cylindrical hole of radius 1 passing through its center. The proposed method involves using integration techniques, specifically revolving a semicircle around the line y=1 to derive the volume. Participants suggest utilizing the concept of "polar caps" and the "washer method" for accurate volume calculation. The integration of the polar caps and the subtraction of the cylinder's volume from the sphere's volume are emphasized as critical steps in solving the problem.

PREREQUISITES
  • Understanding of volume calculation for solids of revolution
  • Familiarity with integration techniques, particularly the washer method
  • Knowledge of polar coordinates and their application in volume problems
  • Basic geometry of spheres and cylinders
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  • Study the washer method for calculating volumes of solids of revolution
  • Learn about integrating polar coordinates to find areas and volumes
  • Explore the concept of polar caps in volume calculations
  • Practice problems involving the volume of spheres with cylindrical holes
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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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