The Length Of A Contour

Question

I have just begun a course in complex analysis and have been presented with this definition: $$\gamma(t):[a,b]\to\Bbb{C}\text{ be differentiable. The length of the curve }\\ \\\{\gamma(t):a\leq t\leq b\}$$$\text{is defined to be}$ $$\int_a^b|\gamma'(t)|dt$$ I would greatly appreciate if anybody could help explain to me why this is the case.

Answer 1

We can approximate the length of the curve using a "polygonal" approximation, adding up the lengths of small line segments such as from $\gamma(t)$ to $\gamma(t+\Delta t)$. But the length of that line segment is $|\gamma(t+\Delta t)-\gamma(t)|=\left|\dfrac{\gamma(t+\Delta t)-\gamma(t)}{\Delta t}\Delta t\right|\approx\left|\gamma'(t)\right|\Delta t$. When we add up all the segments and let $\Delta t$ become small, we approach $\int_a^b\left|\gamma'(t)\right|\,\mathrm d t$.

For a more detailed discussion of length of a smooth curve, see Definition [of arc length] for a smooth curve on Wikipedia. And for an introduction to calculus students, see Arc Length of a Parametric Curve on openstax or Arc Length with Vector Functions from Paul's Online Notes.