Content deleted Content added
→History: Hyperlink added. Tags: Mobile edit Mobile app edit Android app edit App section source |
|||
| (25 intermediate revisions by 11 users not shown) | |||
Line 14:
==History==
[[File:Newgrange Entrance Stone.jpg|thumb|225px|[[Megalithic art]] from [[Newgrange]] showing an early interest in curves]]
Interest in curves began long before they were the subject of mathematical study. This can be seen in numerous examples of their decorative use in art and on everyday objects dating back to prehistoric
times.<ref name="Lockwood">Lockwood p. ix</ref> Curves, or at least their graphical representations, are simple to create, for example with a stick on the sand on a beach.
Line 92:
In particular, the length <math> s </math> of the [[graph of a function|graph]] of a continuously differentiable function <math> y = f(x) </math> defined on a closed interval <math> [a,b] </math> is
:<math>
s = \int_{a}^{b} \sqrt{1 + [f'(x)]^{2}} ~ \mathrm{d}{x}
</math>
which can be thought of intuitively as using the [[Pythagorean theorem]] at the infinitesimal scale continuously over the full length of the curve.<ref>{{Cite book|url=https://books.google.com/books?id=OS4AAAAAYAAJ&dq=length+of+a+curve+formula+pythagorean&pg=RA2-PA108|title=The Calculus|last1=Davis|first1=Ellery W.|last2=Brenke|first2=William C.|date=1913|publisher=MacMillan Company|isbn=9781145891982|page=108|language=en}}</ref>
More generally, if <math> X </math> is a [[metric space]] with metric <math> d </math>, then we can define the length of a curve <math> \gamma: [a,b] \to X </math> by
| |||