Arc Length: Polygonal Approximation, Integral Formulas, and Applications

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What is arc length?

Arc length is the distance measured along a curve between its endpoints, like the length of a string that has been straightened.

What is a polygonal chain?

A path formed by joining a finite sequence of points on a curve, in their consecutive order, with straight line segments.

How is the length of one chord computed in the plane?

For endpoints $(x_i,y_i)$ and $(x_{i+1},y_{i+1})$, use $\sqrt{(x_{i+1}-x_i)^2+(y_{i+1}-y_i)^2}$.

What is chordal distance?

It is the cumulative length of the straight segments in a polygonal chain approximating a curve.

What does rectifiable mean?

A curve is rectifiable if the lengths of all polygonal chains along it are bounded above by a finite number; their least upper bound is its length.

Quiz questions

1. For a continuously differentiable planar curve $(x(t),y(t))$ on $[a,b]$, which integral gives its arc length?Answer: $L=\int_a^b\sqrt{x'(t)^2+y'(t)^2}\,dt$
2. For a suitable reparameterization that traces the same geometric curve, what happens to its arc length?Answer: Its length stays the same, although the rate of travel may change.
3. For a differentiable graph $y=f(x)$ on $[a,b]$, which expression gives its arc length?Answer: $L=\int_a^b\sqrt{1+[f'(x)]^2}\,dx$
4. For a polar curve $r=r(\theta)$, which term accounts for distance traveled as the angle changes?Answer: $r^2$ in the squared-speed expression
5. For a curve on a parameterized surface $\mathbf{x}(u,v)$ with parameter path $(u(t),v(t))$, which expression gives its squared speed?Answer: $g_{11}(u')^2+2g_{12}u'v'+g_{22}(v')^2$