Surface area of an ellipsoid If the top half of the

Chapter 6, Problem 34

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Surface area of an ellipsoid If the top half of the ellipse x2 a2 + y2 b2 = 1 is revolved about the x-axis, the result is an ellipsoid whose axis along the x-axis has length 2a, whose axis along the y-axis has length 2b, and whose axis perpendicular to the xy-plane has length 2b. We assume that 0 6 b 6 a (see figure). Use the following steps to find the surface area S of this ellipsoid. a x b y _ _ 1 x2 a2 y2 b2 a. Use the surface area formula to show that S = 4pb a L a 0 2a2 - c2x2 dx, where c2 = 1 - b2 a2. b. Use the change of variables u = cx to show that S = 4pb 2a2 - b2L 2a2 - b2 0 2a2 - u2 du. c. A table of integrals shows that L 2a2 - u2 du = 1 2 au2a2 - u2 + a2 sin-1 u a b + C. Use this fact to show that the surface area of the ellipsoid is S = 2pbab + a2 2a2 - b2 sin-1 2a2 - b2 a b. d. If a and b have units of length (say, meters), what are the units of S according to this formula? e. Use part (a) to show that if a = b, then S = 4pa2, which is the surface area of a sphere of radius a.