4708.06 to the nearest thousand
2 answers:
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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
The arc length can be computed with a line integral, but first we'll need a parameterization for
Now,
where
Some rewriting:
Complete the square to get
So in the integral, you can substitute
Next substitute
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