Exercise 19

We can adapt the arc length formula to curves in 2-space that define $y$ as a function of $x$ as the following activity shows.

Let $y = f(x)$ define a smooth curve in 2-space. Parameterize this curve and use Equation this equation to show that the length of the curve defined by $f$ on an interval $[a,b]$ is $$\begin{gathered} \int_a^b \sqrt{1+[f'(t)]^2} \, dt. \end{gathered}$$