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There are quite simple, intuitive and straightforward expressions for evaluating the area or volume of a figure. But why is the expression for the length of a curve so complicated?

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In what sense is it more complicated? Is it because of the appearance of the square root which makes it hard to find an anti-derivative, even though the function may have a nice anti-derivative? – Eric O. Korman Jul 27 '10 at 20:54
Yes; the square root makes things more complicated. Moreover in the other cases, it is a differential form of a certain weight. – user218 Jul 27 '10 at 20:56
In all cases you're integrating a differential form. The cases are really not different since in each one you integrate the volume element over a manifold (a 1-manifold in the case of finding the length of the curve). The difference is that in finding the length of the curve, the volume element is induced by the metric. – Eric O. Korman Jul 27 '10 at 21:01

2 Answers 2

The length of a curve doesn't transform in a nice way under scaling of one of the variables the same way that the area of a 2d figure or the volume of a 3d figure does. That puts a hard lower bound on how complicated it has to be.

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Obviously this depends on the shape, as you implied. I would say for a circle the circumference formula is simpler.

Also the length of the curve is a volume in some sense, its just 1 dimensional.

Usually in calculus we are calculating the area relative to the the x-axis, and your formula is in terms of $x$, so that makes things simpler. I imagine that if the curve was parameterized in terms of its arc length finding its length would be quite trivial, and finding the area between the curve and the x-axis would seem very complicated.

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