区分求積法の例

右端型　：${\displaystyle \underset{n\to \infty }{lim}\frac{1}{n}\left\{f\left(\frac{1}{n}\right)+f\left(\frac{2}{n}\right)+\cdots +f\left(\frac{n-1}{n}\right)+f\left(\frac{n}{n}\right)\right\}}$${\displaystyle ={\int }_0^{1}f\left(x\right)dx}$

左端型　：${\displaystyle \underset{n\to \infty }{lim}\frac{1}{n}\left\{f\left(0\right)+f\left(\frac{1}{n}\right)+f\left(\frac{2}{n}\right)+\cdots +f\left(\frac{n-1}{n}\right)\right\}}$${\displaystyle ={\int }_0^{1}f\left(x\right)dx}$

${\displaystyle {\int }_0^1{x}^{2}dx}$${\displaystyle =\underset{n\to \infty }{lim}\frac{1}{n}\left\{f\left(1\right)+f\left(2\right)+\cdots +f\left(n-1\right)+f\left(n\right)\right\}}$

${\displaystyle =\underset{n\to \infty }{lim}\frac{1}{n}\left\{{\left(\frac{1}{n}\right)}^2+{\left(\frac{2}{n}\right)}^2+\cdots +{\left(\frac{n-1}{n}\right)}^2+{\left(\frac{n}{n}\right)}^2\right\}}$

${\displaystyle =\underset{n\to \infty }{lim}\frac{1}{n}·\frac{1}{n^2}\left\{1^2+2^2+\cdots +{\left(n-1\right)}^2+n^2\right\}}$

${\displaystyle =\underset{n\to \infty }{\lim }\frac{1}{n}·\frac{1}{n^2}\left\{{\displaystyle \sum _{k=1}^n{k}^2}\right\}}$

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

${\displaystyle =\underset{n\to \infty }{lim}\frac{1}{n}·\frac{1}{n^2}·\frac{1}{6}n\left(n+1\right)\left(2n+1\right)}$

${\displaystyle =\underset{n\to \infty }{lim}\frac{1}{6}\left(1+\frac{1}{n}\right)\left(2+\frac{1}{n}\right)}$

${\displaystyle n\to \infty }$ のとき，${\displaystyle \frac{1}{n}\to 0}$ より

${\displaystyle =\frac{1}{6}\left(1+0\right)\left(2+0\right)}$

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

${\displaystyle {\int }_0^1{x}^{2}dx}$${\displaystyle =\underset{n\to \infty }{lim}\frac{1}{n}\left\{f\left(0\right)+f\left(1\right)+\cdots +f\left(n-2\right)+f\left(n-1\right)\right\}}$

${\displaystyle =\underset{n\to \infty }{lim}\frac{1}{n}\left\{{\left(\frac{0}{n}\right)}^2+{\left(\frac{1}{n}\right)}^2+\cdots +{\left(\frac{n-2}{n}\right)}^2+{\left(\frac{n-1}{n}\right)}^2\right\}}$

${\displaystyle =\underset{n\to \infty }{\lim }\frac{1}{n}·\frac{1}{n^2}\left\{0^2+1^2+\cdots +{\left(n-2\right)}^2+{\left(n-1\right)}^2\right\}}$

${\displaystyle =\underset{n\to \infty }{\lim }\frac{1}{n}·\frac{1}{n^2}\left\{0^2+{\displaystyle \sum _{k=1}^{n-1}{k}^2}\right\}}$

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

${\displaystyle =\underset{n\to \infty }{lim}\frac{1}{n}·\frac{1}{n^2}·\frac{1}{6}\left(n-1\right)n\left\{2\left(n-1\right)+1\right\}}$

${\displaystyle =\underset{n\to \infty }{lim}\frac{1}{n}·\frac{1}{n^2}·\frac{1}{6}\left(n-1\right)n\left(2n-1\right)}$

${\displaystyle =\underset{n\to \infty }{lim}\frac{1}{6}\left(1-\frac{1}{n}\right)\left(2-\frac{1}{n}\right)}$

${\displaystyle n\to \infty }$ のとき，${\displaystyle \frac{1}{n}\to 0}$ より

${\displaystyle =\frac{1}{6}\left(1-0\right)\left(2-0\right)}$

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

${\displaystyle {\displaystyle \sum _{k=1}^{10}\frac{1}{10}f\left(\frac{k}{10}\right)}}$${\displaystyle =\frac{1}{10}\left\{f\left(\frac{1}{10}\right)+f\left(\frac{2}{10}\right)+\cdots +f\left(\frac{9}{10}\right)+f\left(\frac{10}{10}\right)\right\}}$

${\displaystyle =\frac{1}{10}\left\{{\left(\frac{1}{10}\right)}^2+{\left(\frac{2}{10}\right)}^2+\cdots +{\left(\frac{9}{10}\right)}^2+{\left(\frac{10}{10}\right)}^2\right\}}$

${\displaystyle =\frac{1}{10}\left\{\frac{1}{100}+\frac{4}{100}+\cdots +\frac{81}{100}+\frac{100}{100}\right\}}$

${\displaystyle =\frac{1}{10}·\frac{385}{100}}$

${\displaystyle =0.385}$

${\displaystyle {\displaystyle \sum _{k=1}^{10}\frac{1}{10}f\left(\frac{k-1}{10}\right)}}$${\displaystyle =\frac{1}{10}\left\{f\left(0\right)+f\left(\frac{1}{10}\right)+f\left(\frac{2}{10}\right)+\cdots +f\left(\frac{9}{10}\right)\right\}}$

${\displaystyle =\frac{1}{10}\left\{{\left(0\right)}^2+{\left(\frac{1}{10}\right)}^2+{\left(\frac{2}{10}\right)}^2+\cdots +{\left(\frac{9}{10}\right)}^2\right\}}$

${\displaystyle =\frac{1}{10}\left\{0+\frac{1}{100}+\frac{4}{100}+\cdots +\frac{81}{100}\right\}}$

${\displaystyle =\frac{1}{10}·\frac{285}{100}}$

${\displaystyle =0.285}$

${\displaystyle {\displaystyle \sum _{k=1}^{1000}\frac{1}{1000}f\left(\frac{k}{1000}\right)}}$

${\displaystyle =\frac{1}{1000}\left\{f\left(\frac{1}{1000}\right)+f\left(\frac{2}{1000}\right)+\cdots +f\left(\frac{999}{1000}\right)+f\left(\frac{1000}{1000}\right)\right\}}$

${\displaystyle =\frac{1}{1000}\left\{{\left(\frac{1}{1000}\right)}^2+{\left(\frac{2}{1000}\right)}^2+\cdots +{\left(\frac{999}{1000}\right)}^2+{\left(\frac{1000}{1000}\right)}^2\right\}}$

${\displaystyle =\frac{1}{1000}\left\{\frac{1}{1000000}+\frac{4}{1000000}+\cdots +\frac{998001}{1000000}+\frac{1000000}{1000000}\right\}}$

${\displaystyle =\frac{1}{1000}·\frac{33383350}{1000000}}$

${\displaystyle =0.3338335}$

${\displaystyle {\displaystyle \sum _{k=1}^{1000}\frac{1}{1000}f\left(\frac{k-1}{100}\right)}}$${\displaystyle =\frac{1}{1000}\left\{f\left(0\right)+f\left(\frac{1}{1000}\right)+f\left(\frac{2}{1000}\right)+\cdots +f\left(\frac{999}{1000}\right)\right\}}$

${\displaystyle =\frac{1}{1000}\left\{{\left(0\right)}^2+{\left(\frac{1}{1000}\right)}^2+{\left(\frac{2}{1000}\right)}^2+\cdots +{\left(\frac{999}{1000}\right)}^2\right\}}$

${\displaystyle =\frac{1}{1000}\left\{0+\frac{1}{1000000}+\frac{4}{1000000}+\cdots +\frac{998001}{1000000}\right\}}$

${\displaystyle =\frac{1}{1000}·\frac{33283350}{1000000}}$

${\displaystyle =0.3328335}$