  We begin by studying the surface area of an ellipsoid in n-dimensional
Euclidean space as the function of the lengths of the semi-axes. We write down
an explicit formula as an integral over the unit sphere in n-dimensions and use
this 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 for the surface area and the \emph{isoperimetric ratio} of
an ellipsoid in large dimensions, and to give an expression for the surface in
terms of the Lauricella hypergeometric function. We then write down general
formulas for the volumes of projections of ellipsoids, and use them to extend
the above-mentioned results to give explicit and approximate formulas for the
higher integral mean curvatures of ellipsoids. Some of our results can be
expressed as ISOPERIMETRIC results for higher mean curvatures.
