Negative arc length value. True or not?

Question

I saw a post about this and someone said the arc length is an integral of a positive function, so it is positive. But by solving this exercise I found the arclength as a negative value. The arc length for $ y= ln (1-x^2)$ from x=0 to x=1/2. The result I found is $L= -ln(2)-ln(3/2)+1/2 = -0.5986 $. I calculated it by myself and then I use an online program to calculate it and it gives the same number. So, is it true? Or can't an arc length have a negative value?

Answer 1

Your mistake was in taking the square root of

$$1+\left(\frac{dy}{dx}\right)^2 = \left(\frac{x^2+1}{x^2-1}\right)^2$$

This is $\dfrac{x^2+1}{1-x^2}$, not $\dfrac{x^2+1}{x^2-1}$, because $x^2-1$ is negative for $x\in[0,\frac12]$.

Answer 2

Based on your definition of arc length your first step is wrong. Your function is not a positive function, actually $y=\ln (1 - x^2) \leq 0$ on $(-1,1)$.

So to use your definition of arc length, you need to apply it to $-y$. This is the answer for $y$ as well, since distance and angle between points in the plane is preserved by mirror reflections