= Solution
Differentiating the parametrization gives $dx=a\cos\theta\,d\theta$ and $dy=-b\sin\theta\,d\theta$. Thus the <arc length> element is
$$
ds=\sqrt{a^2\cos^2\theta+b^2\sin^2\theta}\,d\theta
=a\sqrt{1-e^2\sin^2\theta}\,d\theta.
$$
All four quadrants give the same contribution. Consequently the <ellipse> perimeter is
$$
\boxed{L=a\int_0^{2\pi}\sqrt{1-e^2\sin^2\theta}\,d\theta
=4aE(e,\pi/2)}.
$$
The final factor is the <complete elliptic integral of the second kind>, evaluated at the <eccentricity of a conic> $e$.
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