I am asked too calculate the exact length of the intersection curve of the two equations $$ x^2 = 2 y \\ 3z = xy \\ $$
From the point (0,0,0) to the point (6,18,36).
I don't know if I am doing the right thing, but I am starting to isolate $x$ from the first equation and insert it in the second equation to find the equation of the intersection curve.
I find
$$ z = \frac{x^3}{6} $$
The way to compute it is probably to find transform the equation of the intersection into a parametric equation using a new variable t for example. From there, I know how to compute the length, but I will have problem of setting the interval of the integral expressed as the t variable.
Can someone give me a clue on how to passed from an equation of this form to a parametric equation and give me a hint on how to set the interval?
Thank you.
From what you have already found you can use $t=x$ as a parameter:
$$x=t,\ y=t^2/2,\ z=t^3/6,$$
so the interval becomes $[0,6].$