Find the length of the part of the astroid $x^{2/3} + y^{2/3} = 1$ which is contained in the second quadrant (x ≤ 0, y ≥ 0).
The textbook starts the solution by giving the following parameterisation:
First we look for a parameterisation of the astroid. We see that $x^{1/3}$ and $y^{1/3}$ are on a circle of radius $1$, so we have the usual parameterisation $(x^{1/3}, y^{1/3}) = (\cos(t), \sin(t))$. Therefore, we choose the parameterisation $r(t) = (\cos^3(t), > \sin^3(t))$.
I understand that, when parameterising using polar coordinates, we usually set $x = \rho\cos(t)$ and $y = \rho\sin(t)$. However, I do not understand the reasoning behind the parameterisation done in the solution: Why do they use $x^{1/3}$ and $y^{1/3}$ when we have $x^{2/3}$ and $y^{2/3}$ in the equation of the asteroid?
I would greatly appreciate it if people could please take the time to clarify this point.