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# 曲線の長さ

$3\displaystyle{\begin{array}{lll}s& =\int _{{\alpha}}^{{\beta}} \sqrt{ { \( \frac{\hspace{2} dx \hspace{2}}{\hspace{2} dt \hspace{2}} \) }^{2}+{ \( \frac{\hspace{2} dy \hspace{2}}{\hspace{2} dt \hspace{2}} \) }^{2} }dt & \vspace{6}\\ & =\int _{{\alpha}}^{{\beta}} \sqrt{ { \{ {u}^{\prime } \(t\) \} }^{2}+{ \{ {v}^{\prime } \(t\) \} }^{2} }dt & \vspace{6}\\ \end{array}}$

$3\displaystyle{\begin{array}{lll}s& =\int _{a}^{b} \sqrt{ 1+{ \( \frac{\hspace{2} dy \hspace{2}}{\hspace{2} dx \hspace{2}} \) }^{2} }dx & \vspace{6}\\ & =\int _{a}^{b} \sqrt{ 1+{ \{ {f}^{\prime } \(x\) \} }^{2} }dx & \vspace{6}\\ \end{array}}$

となる．ただし，$3\displaystyle{a=u \(\alpha \) }$$3\displaystyle{b=u \({\beta}\) }$ である．

## ■導出

$3\displaystyle{\begin{array}{lll} {\Delta}\hspace{1}{s}_{i} & =\sqrt{ { \( {\Delta}\hspace{1}{x}_{i} \) }^{2}+{ \( {\Delta}\hspace{1}{y}_{i} \) }^{2} } & \vspace{6}\\ & =\sqrt{ { \( \frac{\hspace{2} {\Delta}\hspace{1}{x}_{i} \hspace{2}}{\hspace{2} {\Delta}\hspace{1}{t}_{i} \hspace{2}} \) }^{2}+{ \( \frac{\hspace{2} {\Delta}\hspace{1}{y}_{i} \hspace{2}}{\hspace{2} {\Delta}\hspace{1}{t}_{i} \hspace{2}} \) }^{2} }{\Delta}\hspace{1}{t}_{i} & \vspace{6}\\ \end{array}}$

$3\displaystyle{\begin{array}{lll} { \lim }\limits_{ n\rightarrow \infty }{\sum }\limits_{ i=1 }^{n} \sqrt{ { \( \frac{\hspace{2} {\Delta}\hspace{1}{x}_{i} \hspace{2}}{\hspace{2} {\Delta}\hspace{1}{t}_{i} \hspace{2}} \) }^{2}+{ \( \frac{\hspace{2} {\Delta}\hspace{1}{y}_{i} \hspace{2}}{\hspace{2} {\Delta}\hspace{1}{t}_{i} \hspace{2}} \) }^{2} }{\Delta}\hspace{1}{t}_{i} & =\int _{{\alpha}}^{{\beta}} \sqrt{ { \( \frac{\hspace{2} dx \hspace{2}}{\hspace{2} dt \hspace{2}} \) }^{2}+{ \( \frac{\hspace{2} dy \hspace{2}}{\hspace{2} dt \hspace{2}} \) }^{2} }dt & \vspace{6}\\ & =\int _{{\alpha}}^{{\beta}} \sqrt{ { \{ {u}^{\prime } \(t\) \} }^{2}+{ \{ {v}^{\prime } \(t\) \} }^{2} }dt & \vspace{6}\\ \end{array}}$

となる． 一方

$3\displaystyle{\begin{array}{lll} {\Delta}\hspace{1}{s}_{i} & =\sqrt{ { \( {\Delta}\hspace{1}{x}_{i} \) }^{2}+{ \( {\Delta}\hspace{1}{y}_{i} \) }^{2} } & \vspace{6}\\ & =\sqrt{ 1+{ \( \frac{\hspace{2} {\Delta}\hspace{1}{y}_{i} \hspace{2}}{\hspace{2} {\Delta}\hspace{1}{x}_{i} \hspace{2}} \) }^{2} }{\Delta}\hspace{1}{x}_{i} & \vspace{6}\\ \end{array}}$

と考えると，曲線$3\displaystyle{\text{AB}}$  $3\displaystyle{ \( a\leq x\leq b \) }$ の長さは

$3\displaystyle{\begin{array}{lll} { \lim }\limits_{ n\rightarrow \infty }{\sum }\limits_{ i=1 }^{n} \sqrt{ 1+{ \( \frac{\hspace{2} {\Delta}\hspace{1}{y}_{i} \hspace{2}}{\hspace{2} {\Delta}\hspace{1}{x}_{i} \hspace{2}} \) }^{2} }{\Delta}\hspace{1}{x}_{i} & =\int _{a}^{b} \sqrt{ 1+{ \( \frac{\hspace{2} dy \hspace{2}}{\hspace{2} dx \hspace{2}} \) }^{2} }dx & \vspace{6}\\ & =\int _{a}^{b} \sqrt{ 1+{ \{ {f}^{\prime } \(x\) \} }^{2} }dx & \vspace{6}\\ \end{array}}$

となりる．

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