  Integrals related to the surface area of arbitrary ellipsoids are derived,
evaluated, and compared with each other and existing integrals found in the
literature. We clarify the literature on the ellipsoid area problem, which
dates back to Legendre's original 1811 work. We derive and evaluate several
additional integrals involving or evaluating to elliptic integrals. These new
integrals are natural extensions of those found in the literature and
constitute useful additions to standard integrals compendia.
