Consider a plane curve defined by the parametric equations \(x=x(t)\) and \(y=y(t)\) for \(a \le t \le b\).
Suppose we want to find the arc length of this curve between \(t=a\) and \(t=b\). Let \(x(a) = \alpha\) and \(x(b) = \beta\). Finding the arc length of the curve between \(t=a\) and \(t=b\) is the same as finding the arc length of the curve between \(x = \alpha\) and \(x = \beta\):
\[L = \int_\alpha^\beta \sqrt{1+ \left( \frac{dy}{dx} \right)^2} \ dx\]
The integral with respect to \(t\) would be:
\[\begin{align} L &= \int_a^b \sqrt{1+ \left( \frac{dy/dt}{dx/dt} \right)^2} \frac{dx}{dt}\ dt \\ &= \int_a^b \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \ dt \end{align}\]