calculuscurves
I have a quick question that has been bothering me since I took my calc 2 final earlier. The question dealt with arc length and was pretty much plug and chug using the the arc length formula. My concern is my final answer left me with a negative length with didn't seem right to me, but I couldn't figure out any place where I went wrong.
Is a negative arc length a possible answer?
One way to gain the intuition behind this is to look at what happens in 2 dimensions. Here, rather than surface area and volume, we look at arc length and area under the curve. When we want to find the area under the curve, we estimate using rectangles. This is sufficient to get the area in a limit; one way to see why this is so is that both the error and the estimate are 2-dimensional, and so we aren't missing any extra information.
However, the analogous approach to approximating arc length is obviously bad: this would amount to approximating the curve by a sequence of constant steps (i.e. the top of the rectangles in a Riemann sum) and the length of this approximation is always just the length of the domain. Essentially, we are using a 1-dimensional approximation (i.e. only depending on $x$) for a 2-dimensional object (the curve), and so our approximation isn't taking into account the extra length coming from the other dimension. This is why the arc length is computed using a polygonal approximation by secants to the curve; this approximation incorporates both change in $x$ and change in $y$.
Why is this relevant to solids of revolution? Well, in essence, the volume and surface area formulae are obtained by simply rotating the corresponding 2-dimensional approximation rotated around an axis, and taking a limit. If it wouldn't work in 2 dimensions, it certainly won't work in 3 dimensions.
$y = 2+\frac{2}{x-1}$
$ L = \sqrt{x^2 + (2+\frac{2}{x-1})^2}$
Take the derivative of the function inside the square root and equate it to 0
$\frac{dL}{dx} = (x-1)^{3} - 4 = 0$
$x=4^{1/3} + 1$
Thus $L = \sqrt{(4^{1/3} + 1)^2 + (2+2^{1/3})^2}$
Best Answer
The arc length of a curve cannot be negative, just as the distance between two points cannot be negative. But one can use the usual arc length formula to define an arc length parameterization of a curve, and this parameter can take on negative values (just as one can talk about negative displacements in some coordinate system.)
(The above prohibitions do not hold, by the way, for "indefinite signature metrics", like the Lorentzian spacetimes of general relativity.)