A prolate spheroid is a spheroid that is "pointy" instead of "squashed," i.e., one for which the polar radius is greater than the equatorial radius , so (called "spindle-shaped ellipsoid" by Tietze 1965, p. 27). A symmetrical egg (i.e., with the same shape at both ends) would approximate a prolate spheroid. A prolate spheroid is a surface of revolution obtained by rotating an ellipse about its major axis (Hilbert and Cohn-Vossen 1999, p. 10), and has Cartesian equations
(1)
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The surface area of a prolate spheroid can be computed as a surface of revolution about the z-axis,
(2)
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with radius as a function of given by
(3)
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The integrand is then
(4)
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and the integral is given by
(5)
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(6)
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Using the identity
(7)
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(where the sign of the numerator is flipped from the definition of the eccentricity of an oblate spheroid) then gives
(8)
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(Beyer 1987, p. 131). Note that this is the conventional form in which the surface area of a prolate spheroid is written, although it is formally equivalent to the conventional form for the oblate spheroid via the identity
(9)
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where is defined by
(10)
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