The arc length formula for a curve defined using parametric functions \(x(t)\) and \(y(t)\) between \(t=b\) and \(t=a\) is:
\[ L = \int_a^b \sqrt{x'(t)^2 + y'(t)^2} \ dt \]
Let \(\textbf{r}(t) = y(t)\textbf{i} + x(t)\textbf{j}\). The arc length made traced by \(\textbf{r}(t)\) between \(t=b\) and \(t=a\) is:
\[ s = \int_a^b \sqrt{x'(t)^2 + y'(t)^2} \ dt = \int_a^b \Vert \textbf{r}'(t) \Vert \ dt \]
This is assuming \(\textbf{r}(t)\) defines a smooth curve. If \(\textbf{r}(t)\) is three dimensional:
\[ s = \int_a^b \sqrt{x'(t)^2 + y'(t)^2 + z'(t)^2} \ dt = \int_a^b \Vert \textbf{r}'(t) \Vert \ dt\]
Suppose \(a\) is a constant and \(b\) is a variable, we can define the arc length function as:
\[ s(b) = \int_a^b \sqrt{x'(t)^2 + y'(t)^2 + z'(t)^2} \ dt = \int_a^b \Vert \textbf{r}'(t) \Vert \ dt\]
According to the Fundamental Theorem of Calculus:
\[ \frac{d}{db} s(b) = \Vert \textbf{r}'(b) \Vert \]