Solution
= Solution
Differentiate the <parametric curve>:
$$
\dot x=at\cos t,\qquad \dot y=at\sin t.
$$
The <arc length> element is $ds=\sqrt{\dot x^2+\dot y^2}\,dt=|a|t\,dt$ on $0\le t\le2$. Thus
$$
\boxed{L=\int_0^2|a|t\,dt=2|a|.}
$$
For a positive geometric scale $a$, this is $2a$. The absolute value is necessary when the unspecified real constant $a$ is negative; $a=0$ gives zero <arc length>.