Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2011 / Paper 2 / 16 / b / Solution

Differentiating the parametrization gives $dx=a\cos \theta d\theta$ and $dy=-b\sin \theta d\theta$. Thus the arc length element isAll four quadrants give the same contribution. Consequently the ellipse perimeter isThe final factor is the complete elliptic integral of the second kind, evaluated at the eccentricity of a conic $e$.