I read that, for the arc length formula for curve (f(t),g(t)) to work, the curve must be traced from left to right as t varies from the lower limit to the upper limit, or dx/dt>0 in the interval.
1.What is the reason for this condition?
- They worked out an example <3sint, 3cost> in the interval t€(0,2pi). Clearly, 3sint is not an increasing function in the given interval. However, they simply used the arc length formula as before.
There's no need for any such restriction. The arclength formula comes from taking the length of a piecewise-linear approximation to the curve,
$$\sum_i \sqrt{(\Delta x_i)^2 + (\Delta y_i)^2} = \sum_i \sqrt{\left(\frac{\Delta x_i}{\Delta t}\right)^2 + \left(\frac{\Delta y_i}{\Delta t}\right)^2} \Delta t$$
and refining it so that in the limit we get
$$\int \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt$$
So, first of all, if we think about the geometry we see that it doesn't matter. But we can even see this from the equation without thinking about what it means: everything of interest here is squared, sign can't possibly have any impact on anything!