区分求積法の例

区分求積法の右端型と左端型について， $3$\displaystyle{f $x$ ={x}^{2}}$ として考える．

右端型　： $3$\displaystyle{{ \lim }\limits_{ n\rightarrow \infty }\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}} \{ f $ \frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}} $ +f $ \frac{\hspace{2}2\hspace{2}}{\hspace{2}n\hspace{2}} $ +\cdots +f $ \frac{\hspace{2} n- 1 \hspace{2}}{\hspace{2}n\hspace{2}} $ +f $ \frac{\hspace{2}n\hspace{2}}{\hspace{2}n\hspace{2}} $ \} }$ $3$\displaystyle{=\int _{0}^{1} f $x$ dx }$

左端型　： $3$\displaystyle{{ \lim }\limits_{ n\rightarrow \infty }\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}} \{ f $0$ +f $ \frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}} $ +f $ \frac{\hspace{2}2\hspace{2}}{\hspace{2}n\hspace{2}} $ +\cdots +f $ \frac{\hspace{2} n- 1 \hspace{2}}{\hspace{2}n\hspace{2}} $ \} }$ $3$\displaystyle{=\int _{0}^{1} f $x$ dx }$

ここで， $3$n$ は積分範囲を分割した数である．

■導出

●定積分の公式を用いた場合

定積分の基本式より

$3$\displaystyle{\int _{a}^{b} f $x$ dx= \[ F $x$ \] _{a}^{b} =F $b$ - F $a$ }$ 　　　（ $3$\displaystyle{F $x$ }$ は $3$\displaystyle{f $x$ }$ の原始関数の1である）

$3$\displaystyle{\int _{0}^{1} {x}^{2}dx }$ $3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}3\hspace{2}} \[ {x}^{3} \] _{0}^{1}}$ $3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}3\hspace{2}}}$ $3$\displaystyle{=0.333\cdots }$

●右端型の区分求積の場合

$3$\displaystyle{\int _{0}^{1} {x}^{2} dx}$ $3$\displaystyle{={ \lim }\limits_{ n\rightarrow \infty }\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}} \{ f $1$ +f $2$ +\cdots +f $ n- 1 $ +f $n$ \} }$

$3$\displaystyle{={ \lim }\limits_{ n\rightarrow \infty }\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}} \{ { $ \frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}} $ }^{2}+{ $ \frac{\hspace{2}2\hspace{2}}{\hspace{2}n\hspace{2}} $ }^{2}+\cdots +{ $ \frac{\hspace{2} n- 1 \hspace{2}}{\hspace{2}n\hspace{2}} $ }^{2}+{ $ \frac{\hspace{2}n\hspace{2}}{\hspace{2}n\hspace{2}} $ }^{2} \} }$

$3$\displaystyle{={ \lim }\limits_{ n\rightarrow \infty }\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}\cdot \frac{\hspace{2}1\hspace{2}}{\hspace{2} {n}^{2} \hspace{2}} \{ {1}^{2}+{2}^{2}+\cdots +{ $ n- 1 $ }^{2}+{n}^{2} \} }$

$3$\displaystyle{={ \lim }\limits_{ n\rightarrow \infty }\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}\cdot \frac{\hspace{2}1\hspace{2}}{\hspace{2} {n}^{2} \hspace{2}}\left{{ \displaystyle{{\sum }\limits_{ k=1 }^{n} {k}^{2} } }\right}}$

$3$\displaystyle{\displaystyle{{\sum }\limits_{ k=1 }^{n} {k}^{2} }=\frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}n\left({ n+1 }\right)\left({ 2n+1 }\right)}$ より

$3$\displaystyle{={ \lim }\limits_{ n\rightarrow \infty }\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}\cdot \frac{\hspace{2}1\hspace{2}}{\hspace{2} {n}^{2} \hspace{2}}\cdot \frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}n $ n+1 $ $ 2n+1 $ }$

$3$\displaystyle{={ \lim }\limits_{ n\rightarrow \infty }\frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}} $ 1+\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}} $ $ 2+\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}} $ }$

$3$\displaystyle{n\rightarrow \infty}$ のとき， $3$\displaystyle{\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}\rightarrow 0}$ より

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}} $ 1+0 $ $ 2+0 $ }$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}3\hspace{2}}}$

●左端型の区分求積の場合

$3$\displaystyle{\int _{0}^{1} {x}^{2} dx}$ $3$\displaystyle{={ \lim }\limits_{ n\rightarrow \infty }\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}} \{ f $0$ +f $1$ +\cdots +f $ n- 2 $ +f $ n- 1 $ \} }$

$3$\displaystyle{={ \lim }\limits_{ n\rightarrow \infty }\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}} \{ { $ \frac{\hspace{2}0\hspace{2}}{\hspace{2}n\hspace{2}} $ }^{2}+{ $ \frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}} $ }^{2}+\cdots +{ $ \frac{\hspace{2} n- 2 \hspace{2}}{\hspace{2}n\hspace{2}} $ }^{2}+{ $ \frac{\hspace{2} n- 1 \hspace{2}}{\hspace{2}n\hspace{2}} $ }^{2} \} }$

$3$\displaystyle{ ={ \lim }\limits_{ n\rightarrow \infty }\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}\cdot \frac{\hspace{2}1\hspace{2}}{\hspace{2} {n}^{2} \hspace{2}}\left{{ {0}^{2}+{1}^{2}+\cdots +{ \left({ n- 2 }\right) }^{2}+{ \left({ n- 1 }\right) }^{2} }\right} }$

$3$\displaystyle{={ \lim }\limits_{ n\rightarrow \infty }\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}\cdot \frac{\hspace{2}1\hspace{2}}{\hspace{2} {n}^{2} \hspace{2}}\left{{ {0}^{2}+\displaystyle{{\sum }\limits_{ k=1 }^{ n- 1 } {k}^{2} } }\right}}$

$3$\displaystyle{\displaystyle{{\sum }\limits_{ k=1 }^{ n- 1 } {k}^{2} }=\frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}\left({ n- 1 }\right)n\left{{ 2\left({ n- 1 }\right)+1 }\right}}$ より（ここを参照）

$3$\displaystyle{={ \lim }\limits_{ n\rightarrow \infty }\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}\cdot \frac{\hspace{2}1\hspace{2}}{\hspace{2} {n}^{2} \hspace{2}}\cdot \frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}} $ n- 1 $ n \{ 2 $ n- 1 $ +1 \} }$

$3$\displaystyle{={ \lim }\limits_{ n\rightarrow \infty }\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}\cdot \frac{\hspace{2}1\hspace{2}}{\hspace{2} {n}^{2} \hspace{2}}\cdot \frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}} $ n- 1 $ n $ 2n- 1 $ }$

$3$\displaystyle{={ \lim }\limits_{ n\rightarrow \infty }\frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}} $ 1- \frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}} $ $ 2- \frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}} $ }$

$3$\displaystyle{n\rightarrow \infty}$ のとき， $3$\displaystyle{\frac{\hspace{2}1\hspace{2}}{\hspace{2}n\hspace{2}}\rightarrow 0}$ より

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}} $ 1- 0 $ $ 2- 0 $ }$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}3\hspace{2}}}$

よって，右端型と左端方のどちらを用いても，定積分の公式を用いた場合の答と同じ値を得ることができる．

■具体的な $3$n$ の値における計算結果

●その1

$3$\displaystyle{n=10}$ の場合

右端型

$3$\displaystyle{\displaystyle{{\sum }\limits_{ k=1 }^{ 10 } \frac{\hspace{2}1\hspace{2}}{\hspace{2} 10 \hspace{2}}f $ \frac{\hspace{2}k\hspace{2}}{\hspace{2} 10 \hspace{2}} $ }}$ $3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2} 10 \hspace{2}} \{ f $ \frac{\hspace{2}1\hspace{2}}{\hspace{2} 10 \hspace{2}} $ +f $ \frac{\hspace{2}2\hspace{2}}{\hspace{2} 10 \hspace{2}} $ +\cdots +f $ \frac{\hspace{2}9\hspace{2}}{\hspace{2} 10 \hspace{2}} $ +f $ \frac{\hspace{2} 10 \hspace{2}}{\hspace{2} 10 \hspace{2}} $ \} }$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2} 10 \hspace{2}} \{ { $ \frac{\hspace{2}1\hspace{2}}{\hspace{2} 10 \hspace{2}} $ }^{2}+{ $ \frac{\hspace{2}2\hspace{2}}{\hspace{2} 10 \hspace{2}} $ }^{2}+\cdots +{ $ \frac{\hspace{2}9\hspace{2}}{\hspace{2} 10 \hspace{2}} $ }^{2}+{ $ \frac{\hspace{2} 10 \hspace{2}}{\hspace{2} 10 \hspace{2}} $ }^{2} \} }$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2} 10 \hspace{2}} \{ \frac{\hspace{2}1\hspace{2}}{\hspace{2} 100 \hspace{2}}+\frac{\hspace{2}4\hspace{2}}{\hspace{2} 100 \hspace{2}}+\cdots +\frac{\hspace{2} 81 \hspace{2}}{\hspace{2} 100 \hspace{2}}+\frac{\hspace{2} 100 \hspace{2}}{\hspace{2} 100 \hspace{2}} \} }$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2} 10 \hspace{2}}\cdot \frac{\hspace{2} 385 \hspace{2}}{\hspace{2} 100 \hspace{2}}}$

$3$\displaystyle{=0.385}$

左端型

$3$\displaystyle{\displaystyle{{\sum }\limits_{ k=1 }^{ 10 } \frac{\hspace{2}1\hspace{2}}{\hspace{2} 10 \hspace{2}}f $ \frac{\hspace{2} k- 1 \hspace{2}}{\hspace{2} 10 \hspace{2}} $ }}$ $3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2} 10 \hspace{2}} \{ f $0$ +f $ \frac{\hspace{2}1\hspace{2}}{\hspace{2} 10 \hspace{2}} $ +f $ \frac{\hspace{2}2\hspace{2}}{\hspace{2} 10 \hspace{2}} $ +\cdots +f $ \frac{\hspace{2}9\hspace{2}}{\hspace{2} 10 \hspace{2}} $ \} }$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2} 10 \hspace{2}} \{ { $0$ }^{2}+{ $ \frac{\hspace{2}1\hspace{2}}{\hspace{2} 10 \hspace{2}} $ }^{2}+{ $ \frac{\hspace{2}2\hspace{2}}{\hspace{2} 10 \hspace{2}} $ }^{2}+\cdots +{ $ \frac{\hspace{2}9\hspace{2}}{\hspace{2} 10 \hspace{2}} $ }^{2} \} }$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2} 10 \hspace{2}} \{ 0+\frac{\hspace{2}1\hspace{2}}{\hspace{2} 100 \hspace{2}}+\frac{\hspace{2}4\hspace{2}}{\hspace{2} 100 \hspace{2}}+\cdots +\frac{\hspace{2} 81 \hspace{2}}{\hspace{2} 100 \hspace{2}} \} }$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2} 10 \hspace{2}}\cdot \frac{\hspace{2} 285 \hspace{2}}{\hspace{2} 100 \hspace{2}}}$

$3$\displaystyle{=0.285}$

●その2

$3$\displaystyle{n=1000}$ の場合

右端型

$3$\displaystyle{\displaystyle{{\sum }\limits_{ k=1 }^{ 1000 } \frac{\hspace{2}1\hspace{2}}{\hspace{2} 1000 \hspace{2}}f $ \frac{\hspace{2}k\hspace{2}}{\hspace{2} 1000 \hspace{2}} $ }}$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2} 1000 \hspace{2}} \{ f $ \frac{\hspace{2}1\hspace{2}}{\hspace{2} 1000 \hspace{2}} $ +f $ \frac{\hspace{2}2\hspace{2}}{\hspace{2} 1000 \hspace{2}} $ +\cdots +f $ \frac{\hspace{2} 999 \hspace{2}}{\hspace{2} 1000 \hspace{2}} $ +f $ \frac{\hspace{2} 1000 \hspace{2}}{\hspace{2} 1000 \hspace{2}} $ \} }$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2} 1000 \hspace{2}} \{ { $ \frac{\hspace{2}1\hspace{2}}{\hspace{2} 1000 \hspace{2}} $ }^{2}+{ $ \frac{\hspace{2}2\hspace{2}}{\hspace{2} 1000 \hspace{2}} $ }^{2}+\cdots +{ $ \frac{\hspace{2} 999 \hspace{2}}{\hspace{2} 1000 \hspace{2}} $ }^{2}+{ $ \frac{\hspace{2} 1000 \hspace{2}}{\hspace{2} 1000 \hspace{2}} $ }^{2} \} }$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2} 1000 \hspace{2}} \{ \frac{\hspace{2}1\hspace{2}}{\hspace{2} 1000000 \hspace{2}}+\frac{\hspace{2}4\hspace{2}}{\hspace{2} 1000000 \hspace{2}}+\cdots +\frac{\hspace{2} 998001 \hspace{2}}{\hspace{2} 1000000 \hspace{2}}+\frac{\hspace{2} 1000000 \hspace{2}}{\hspace{2} 1000000 \hspace{2}} \} }$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2} 1000 \hspace{2}}\cdot \frac{\hspace{2} 33383350 \hspace{2}}{\hspace{2} 1000000 \hspace{2}}}$

$3$\displaystyle{=0.3338335}$

左端型

$3$\displaystyle{\displaystyle{{\sum }\limits_{ k=1 }^{ 1000 } \frac{\hspace{2}1\hspace{2}}{\hspace{2} 1000 \hspace{2}}f $ \frac{\hspace{2} k- 1 \hspace{2}}{\hspace{2} 100 \hspace{2}} $ }}$ $3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2} 1000 \hspace{2}} \{ f $0$ +f $ \frac{\hspace{2}1\hspace{2}}{\hspace{2} 1000 \hspace{2}} $ +f $ \frac{\hspace{2}2\hspace{2}}{\hspace{2} 1000 \hspace{2}} $ +\cdots +f $ \frac{\hspace{2} 999 \hspace{2}}{\hspace{2} 1000 \hspace{2}} $ \} }$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2} 1000 \hspace{2}} \{ { $0$ }^{2}+{ $ \frac{\hspace{2}1\hspace{2}}{\hspace{2} 1000 \hspace{2}} $ }^{2}+{ $ \frac{\hspace{2}2\hspace{2}}{\hspace{2} 1000 \hspace{2}} $ }^{2}+\cdots +{ $ \frac{\hspace{2} 999 \hspace{2}}{\hspace{2} 1000 \hspace{2}} $ }^{2} \} }$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2} 1000 \hspace{2}} \{ 0+\frac{\hspace{2}1\hspace{2}}{\hspace{2} 1000000 \hspace{2}}+\frac{\hspace{2}4\hspace{2}}{\hspace{2} 1000000 \hspace{2}}+\cdots +\frac{\hspace{2} 998001 \hspace{2}}{\hspace{2} 1000000 \hspace{2}} \} }$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2} 1000 \hspace{2}}\cdot \frac{\hspace{2} 33283350 \hspace{2}}{\hspace{2} 1000000 \hspace{2}}}$

$3$\displaystyle{=0.3328335}$

また，各分割数で計算した場合の値は以下のようになる．（一部の値には誤差を含む）

$3$n$	10	100	1000	10000
右端型	0.385	0.33835	0.3338335	0.333383334999998
左端型	0.285	0.32835	0.3328335	0.333283334999998
$3$n$	20000	50000	100000	1000000
右端型	0.333358333749998	0.3333433334	0.33333833335	0.3333338333335
左端型	0.333308333749998	0.3333233334	0.33332833335	0.3333328333335

この表から，区分求積法は分割数が増えるごとに定積分の公式を用いた場合の値に近づいていくことがわかる．

ホーム>>カテゴリー分類>>積分>>区分求積法の基本式>>区分求積法の例

最終更新日： 2025年6月12日