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Use Polar Coordinates to Find the Volume of a Ring-Shaped Solid

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A cylindrical drill with radius 2 is used to bore a hole throught the center of a sphere of radius 7. Find the volume of the ring shaped solid that remains using polar coordinates.

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Polar coordinates are used to find the volume of a Ring-Shaped solid. The solution is detailed and well presented. The response received a rating of "5" from the student who originally posted the question.

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The figure above is a simple version of how the solid looks. The hole is drilled from top to bottom. Let the radius of the sphere = W and that of the drill = w.

In order to simplify things, align the center of the sphere with the origin (0,0,0) and the axis of the cylindrical hole is aligned with the z-axis, so that the equation of the sphere is
x2 + y2 + z2 = W2 and the equation of the cylindrical hole is x2 + y2 = w2.

So if you look at the xy-plane (i.e. z = 0), we ...

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