Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics IV / 2 / c / Solution

Let $D$ be the filled projected ellipse. The part of the plane bounded by $C$ is the parametrized surfaceIts tangent vectors satisfyThe surface area is thereforeThis also illustrates the orthogonal projection area formula: the unit normal to the plane has vertical component $1/\sqrt{3}$, so projection multiplies area by $1/\sqrt{3}$.