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Math Help - Volume inside a sphere and outside a cylinder

  1. #1
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    Volume inside a sphere and outside a cylinder

    Find the volume that is inside the sphere x^2+y^2+z^2=16 and outside the cylinder x^2+y^2=4 without using integrals. The only information needed is the length of the interior wall that's left in the sphere after the core has been drilled out and one (or more) common geometric formula(s).

    1. Find the volume remaining in a sphere of radius R after a core has been removed leaving an interior wall with a length of 8 units without using any integrals?
    2. Generalize your answer above when the interior wall has a length of A units?

    Not really sure where to start with this, I know how to do it with integrals but I'm completely lost on how to do it without integrals. Any help is greatly appreciated.
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  2. #2
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    Well, presumably you know that the volume of a sphere is given by \frac{4}{3}\pi R^3 and that of a cylinder by \pr r^2 h. Here, you are told that R= 4 and that r= 2. To find the length of the cylinder, imagine lines from the center of the sphere to where the cylinder intercepts the sphere, which is a radius of the sphere and so has length R, and a line from the center of the sphere perpendicular to the axis of the cylinder, which is a radius of the cylinder and so has length r. Those two lines from the hypotenuse and one leg of a right triangle with the half the length of the cylinder the other leg. By the Pythagorean theorem, r^2+ \left(\frac{L}{2}\right)^2= R^2.

    Now, at this point, you have to be careful. The "volume inside the sphere and outside the cylinder" is NOT the volume of the sphere minus the volume of the cylinder. That ignores the two "caps" at each end of the cylinder that will also be removed from the sphere but are beyond the length of the cylinder calculated above. Wikipedia gives a formula for finding the volume of such "spherical caps": Spherical cap - Wikipedia, the free encyclopedia
    Last edited by HallsofIvy; November 23rd 2010 at 04:30 AM.
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