  We study the surface area of an ellipsoid in n-dimensional Euclidean space as
the function of the lengths of their major semi-axes. We write down an explicit
formula as an integral over the unit sphere, use the formula to derive
convexity properties of the surface area, to give sharp estimates for the
surface area of a large-dimensional ellipsoid, to produce asymptotic formulas
in large dimensions, and to give an expression for the surface in terms of the
Lauricella hypergeometric function.
