On the surface area of an ellipsoid and related integrals of elliptic integrals (Q1334860)

The paper deals with integrals of an arbitrary field \(F(x,y,z)\) on the surface area of an ellipsoid \[ S:= \biggl\{ (x,y,z)\in \mathbb{R}^ 3 \mid {{x^ 2} \over {a^ 2}} + {{y^ 2} \over {b^ 2}} + {{z^ 2} \over {c^ 2}} =1 \biggr\}. \] Different orientations of the coordinate axes lead to different (equivalent) integral expressions. Specific choices of the field \(F(x,y,z)\) enable the author to evaluate (or to derive various identities among) integrals containing complete elliptic integrals (of the second kind) as their kernels. As an application the surface area of an arbitrary ellipsoid can be obtained explicitly.