= Scaled length of an Archimedean spiral
{title2=$\lim_{a\to0^+}aL_a=R^2/2$}
Let an <Archimedean spiral> have <polar radius> $r=at$ and <angle> $t$ in <polar coordinates>, with $0\leq t\leq R/a$, where $R>0$ is fixed. Parameterizing by $r$ gives the <arc length>
$$
aL_a=\int_0^R\sqrt{a^2+r^2}\,dr.
$$
The bounds $r\leq\sqrt{a^2+r^2}\leq r+a$ yield
$$
\frac{R^2}{2}\leq aL_a\leq\frac{R^2}{2}+aR.
$$
The <squeeze theorem> proves the displayed <limit>. Thus the <arc length> grows like $R^2/(2a)$ as the turns become closely spaced, without requiring an explicit antiderivative.
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