Let C be the curve of intersection of the parabolic cylinder x2 = 2y, and the surface 3z = xy. Find the exact length of C from the origin to the point (4,8,32/3).

Answer 1

let x=t
then y = t^2/2
z = xy/3 = t^3/6

then the arc length of the curve is

∫[0,4] √(1+t^2+t^4/4) dt = 44/3