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 (4, 8, 32/3)

I've attached a plot of the intersection (highlighted in red) between the parabolic cylinder (orange) and the hyperbolic paraboloid (blue).

The arc length can be computed with a line integral, but first we'll need a parameterization for . This is easy enough to do. First fix any one variable. For convenience, choose .

Now, , and . The intersection is thus parameterized by the vector-valued function



where . The arc length is computed with the integral



Some rewriting:



Complete the square to get



So in the integral, you can substitute to get



Next substitute , so that the integral becomes



This is a fairly standard integral (it even has its own Wiki page, if you're not familiar with the derivation):



So the arc length is