The curve formed by the intersection of a cylinder and a sphere is known as Viviani's curve.
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The problem of finding the lateral surface area of a cylinder of radius internally tangent to a sphere of radius was given in a Sangaku problem from 1825.
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The easiest way to determine the solution is to solve the simultaneous equations
(1)
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(2)
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for and ,
(3)
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(4)
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These give the parametric equations for Viviani's curve in this case (left figure). The surface area can then be found by constructing a series of curved segments (right figure). The arc length element around the surface of the cylinder at a height is given by
(5)
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(6)
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The surface area of one quarter of the surface is then
(7)
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(8)
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(9)
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where some care is needed treating the lower limit,
(10)
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(11)
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The total surface area is then
(12)
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a result obtained in a more roundabout geometric arguments by Rothman (1998). (Note that the answer printed in the original Rothman article was incorrect; the corrected answer has been posted on the Internet version of the article.)