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Volume of a drilled space in a sphere

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I have enclosed all details on the attached file. How do I find the volume of the drilled area?
Consider a sphere of radius R and circular hole 2d drilled through along axis of symmetry...

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Computing volume is a good application for an integral. In this problem, we have symmetry about the axis of the center of the hole, so it makes sense to define cylindrical shells centered on that axis, and ...

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