How do I compute the length of the following curve?

Question

I have the following problem. I need to compute the length of the curve given by

$z^2=2x,3y=xz$ between $(0,0,0)$ and $(\frac{1}{2},\frac{1}{6},1)$

I first want to find my curve, so geometrically I could draw it with geogebra but I can't see how to find my curve. Could maybe someone give me a hint?

Thanks a lot

Answer 1

Hint: Use the fact that $x=\frac{z^2}2$ and that $y=\frac{xz}3=\frac{z^3}6$.

Answer 2

Parametrize your curve $\vec{r}(t) = (x(t), y(t), z(t))$, and then compute the arc-length integral of the curve:

$$ \int_{t_{1}}^{t_{2}} |\vec{r}'(t)| dt $$

Hint: Define $y$, $z$ in terms of $x \in [0, 1/2]$