Let C be the curve of intersection of the parabolic cylinder x^2 = 2y and the surface 3z = xy. Find the exact length of C from the origin to the point (6,18,36)

Question

Let C be the curve of intersection of the parabolic cylinder x^2 = 2y and the surface 3z = xy.  Find the exact length of C from the origin to the point (6,18,36)

turingtest · Answer

this seems quite hard to find a formula for the intersection of the two shapes
I'll be thinking about it

turingtest · Answer

z=x^3/6 but we need it parametric I suppose

turingtest · Answer

y=x^2/2
perhaps
r(t)=(1,t^2/2,t^3/3)

turingtest · Answer

r(t)=(t,t^2/2,t^3/3) I mean

anonymous · Answer

Well the problem I have is finding the integral below. $$\int\limits_{}^{}\sqrt{1+t^2 + \frac{ t^4 }{ 4 }}$$

turingtest · Answer

right, maybe complete the square and trig sub ?

turingtest · Answer

r(t)=(t,t^2/2,t^3/6) I should have written earlier, which leads to your arc length formula...

turingtest · Answer

$$\sqrt{(\frac{t^2}2+1)^2}=(\frac{t^2}2+1)$$

turingtest · Answer

there we go, right?

turingtest · Answer

haha it was already a perfect square :)

turingtest · Answer

@jarobins did that make sense, are you good to go?

anonymous · Answer

Well, I think I'm just needing to brush up on my integration. It does make sense. Thanks