= Solution
Rewrite the integrand as $\sqrt{1-t\cos^2\theta}$. The <binomial series>, uniformly valid in $\theta$ for $|t|$ bounded below one, gives
$$
\sqrt{1-t\cos^2\theta}=1-\frac t2\cos^2\theta-\frac{t^2}8\cos^4\theta+O(t^3).
$$
The integrals of $1,\cos^2\theta,\cos^4\theta$ over a full period are respectively $2\pi,\pi,3\pi/4$. Termwise integration therefore yields the <ellipse perimeter expansion near a circle>
$$
\boxed{y=2\pi-\frac{\pi}{2}t-\frac{3\pi}{32}t^2+O(t^3)
=2\pi\left(1-\frac t4-\frac{3t^2}{64}+O(t^3)\right).}
$$
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