 Solution

The arc length of this parametric curve is the integral of its speed. Differentiation gives $x^{\prime }(t)=at\cos t$ and $y^{\prime }(t)=at\sin t$, soFor a positive scale parameter this is $a/2$. The absolute value is necessary when $a$ is allowed to be any real constant; $a=0$ gives a degenerate curve of zero length. The nondegenerate curve is an involute of a circle.