Past exam of the mathematical methods II course of the University of Oxford / 2021 resit / 4 / Solution

Differentiate the parametric curve:The arc length element is $ds=\sqrt{{\dot{x}}^2+{\dot{y}}^2}dt=|a|tdt$ on $0\le t\le 2$. ThusFor 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.