Pisteiden välisten matkojen summa antaa approksimaation käyrän pituudesta. Käyrän pituus , s , funktiolle f saadaan integraalina
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{\displaystyle \displaystyle s=\int _{a}^{b}{\sqrt {1+[f'(x)]^{2}}}\,dx}
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Olkoon funktio f määritelty suljetulla välillä
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{\displaystyle [a,b]}
f :n rajoittuma tälle välille eli
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{\displaystyle g:[a,b]\to \mathbb {R} }
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{\displaystyle g(x)=f(x)}
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{\displaystyle x\in [a,b]}
f on jatkuva derivaatta f' . Olkoon K funktion g kuvaaja.
Määritellään piste Pi K
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{\displaystyle x_{i},f(x_{i}))}
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{\displaystyle x_{0}<x_{1}<\ldots <x_{n}}
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{\displaystyle x_{0}=a,x_{n}=b}
Tällöin kuvaajan K pituus on peräkkäisten pisteiden
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{\displaystyle (P_{i},P_{i+1})}
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{\displaystyle i\in [0,n-1]}
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{\displaystyle {\begin{aligned}\sum _{i=1}^{n}|{P_{i-1}P_{i}}|&=\sum _{i=1}^{n}{\sqrt {(x_{i}-x_{i-1})^{2}+[f(x_{i})-f(x_{i-1})]^{2}}}\\&=\sum _{i=1}^{n}{\sqrt {\left(1+{\frac {[f(x_{i})-f(x_{i-1})]^{2}}{\Delta x^{2}}}\right)\Delta x^{2}}},\,\Delta x=(x_{i}-x_{i-1})\\&=\sum _{i=1}^{n}{\sqrt {\left(1+\left({\frac {f(x_{i})-f(x_{i-1})}{\Delta x}}\right)^{2}\right)}}\Delta x\\\end{aligned}}}
Kun Δx → 0 , termi
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{\displaystyle {\frac {f(x_{i})-f(x_{i-1})}{\Delta x}}=f'(x)}
Saadaan integraali:
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{\displaystyle L=\int _{a}^{b}{\sqrt {1+[f'(x)]^{2}}}\,dx}
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{\displaystyle {\begin{aligned}(ds)^{2}&{}=(dx)^{2}+(dy)^{2}\\L&{}=\int \ ds\\&{}=\int \ {\sqrt {(dx)^{2}+(dy)^{2}}}\\&{}=\int \ {\sqrt {(dx)^{2}\left[{1+\left({dy \over dx}\right)^{2}}\right]}}\\L&{}=\int _{}^{}{\sqrt {1+\left({dy \over dx}\right)^{2}}}dx\end{aligned}}}
↑ Harjulehto, Petteri & Klén, Riku & Koskenoja, Mika: Analyysiä reaaliluvuilla , s. 192. Helsinki: Unigrafia, 2014. ISBN 978-952-93-4162-7
![{\displaystyle \displaystyle s=\int _{a}^{b}{\sqrt {1+[f'(x)]^{2}}}\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad9852e718e4637132f70e1bcd0adf3a12b51f00)
![{\displaystyle [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935)
![{\displaystyle g:[a,b]\to \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b1d5103d0e36767a27715bcfc57137119294aad)
![{\displaystyle x\in [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/026357b404ee584c475579fb2302a4e9881b8cce)
![{\displaystyle i\in [0,n-1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6186289d8e39bb48f284bfd8d37fa7e8c7470f34)
![{\displaystyle {\begin{aligned}\sum _{i=1}^{n}|{P_{i-1}P_{i}}|&=\sum _{i=1}^{n}{\sqrt {(x_{i}-x_{i-1})^{2}+[f(x_{i})-f(x_{i-1})]^{2}}}\\&=\sum _{i=1}^{n}{\sqrt {\left(1+{\frac {[f(x_{i})-f(x_{i-1})]^{2}}{\Delta x^{2}}}\right)\Delta x^{2}}},\,\Delta x=(x_{i}-x_{i-1})\\&=\sum _{i=1}^{n}{\sqrt {\left(1+\left({\frac {f(x_{i})-f(x_{i-1})}{\Delta x}}\right)^{2}\right)}}\Delta x\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d019365551efd6a1cd0236b88e0438b040d623de)
![{\displaystyle L=\int _{a}^{b}{\sqrt {1+[f'(x)]^{2}}}\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba11f9165247b7a511e42c3451d1f56712237ef4)
![{\displaystyle {\begin{aligned}(ds)^{2}&{}=(dx)^{2}+(dy)^{2}\\L&{}=\int \ ds\\&{}=\int \ {\sqrt {(dx)^{2}+(dy)^{2}}}\\&{}=\int \ {\sqrt {(dx)^{2}\left[{1+\left({dy \over dx}\right)^{2}}\right]}}\\L&{}=\int _{}^{}{\sqrt {1+\left({dy \over dx}\right)^{2}}}dx\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a110a8da8482b55b33aa50406acb203ec97d818)