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$\mathsf{\text{区分求積法}}$

$\stackrel{{\displaystyle {\int }_0^1}f(x)dx}{\mathsf{\text{定積分の定義式}}}$$\Leftarrow$ $\stackrel{{\mathit{l}\mathit{i}\mathit{m}}_{\stackrel{{}^{}}{n\to \mathrm{\infty }}}{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}f\left({\displaystyle \frac{i}{n}}\right)}{\mathsf{\text{区分求積法}}(\mathsf{\text{定積分の定義式の構造}})}$

${\displaystyle \frac{1}{n}}{\displaystyle \sum _{i={\displaystyle \frac{1}{2}}n}^{{\displaystyle \frac{7}{8}}n}}co{s}^2\left({\displaystyle \frac{i\pi }{n}}\right)$$\Rightarrow \left\{\begin{array}{l}\mathsf{\text{級数を表すために、初期値}}{\displaystyle \frac{n}{2}}\mathsf{\text{ と 終了値}}{\displaystyle \frac{7}{8}n}\mathsf{\text{ の分母を通分}}(\mathsf{\text{同じ数に}})\mathsf{\text{する。}}\\ \mathsf{\text{とりあえず分母を}}400\mathsf{\text{にすると、}}\end{array}\right\}$

$\Rightarrow {\displaystyle \frac{1}{n}}{\displaystyle \sum _{i={\displaystyle \frac{200}{400}}n}^{{\displaystyle \frac{350}{400}}n}}co{s}^2\left({\displaystyle \frac{i\pi }{n}}\right)$$={\displaystyle \frac{1}{n}}\{co{s}^2\left({\displaystyle \frac{{\displaystyle \frac{200n\pi }{400}}}{n}}\right)+co{s}^2\left({\displaystyle \frac{{\displaystyle \frac{201n\pi }{400}}}{n}}\right)+co{s}^2\left({\displaystyle \frac{{\displaystyle \frac{202n\pi }{400}}}{n}}\right)+$$\cdots \cdots +co{s}^2\left({\displaystyle \frac{{\displaystyle \frac{349n\pi }{400}}}{n}}\right)+co{s}^2\left({\displaystyle \frac{{\displaystyle \frac{350n\pi }{400}}}{n}}\right)$$\}$

${\displaystyle \frac{1}{n}}{\displaystyle \sum _{i={\displaystyle \frac{1}{2}}n}^{{\displaystyle \frac{7}{8}}n}}co{s}^2\left({\displaystyle \frac{i\pi }{n}}\right)$$\Rightarrow \left\{\mathsf{\text{通分分母を}}10000\mathsf{\text{にすると、}}\right\}\Rightarrow$${\displaystyle \frac{1}{n}}{\displaystyle \sum _{i={\displaystyle \frac{5000}{10000}}n}^{{\displaystyle \frac{1250}{10000}}n}}co{s}^2\left({\displaystyle \frac{i\pi }{n}}\right)$

$={\displaystyle \frac{1}{n}}\{co{s}^2\left({\displaystyle \frac{{\displaystyle \frac{5001n\pi }{10000}}}{n}}\right)+co{s}^2\left({\displaystyle \frac{{\displaystyle \frac{5002n\pi }{10000}}}{n}}\right)+co{s}^2\left({\displaystyle \frac{{\displaystyle \frac{5003n\pi }{10000}}}{n}}\right)+$$\cdots \cdots +co{s}^2\left({\displaystyle \frac{{\displaystyle \frac{1249n\pi }{10000}}}{n}}\right)+co{s}^2\left({\displaystyle \frac{{\displaystyle \frac{1250n\pi }{10000}}}{n}}\right)$$\}$

初項 $co{s}^2\left({\displaystyle \frac{{\displaystyle \frac{n\pi }{2}}}{n}}\right)$$=co{s}^2\left({\displaystyle \frac{n\pi }{2n}}\right)$$=co{s}^2\left({\displaystyle \frac{\pi }{2}}\right)$

末項 $co{s}^2\left({\displaystyle \frac{{\displaystyle \frac{7n\pi }{8}}}{n}}\right)$$=co{s}^2\left({\displaystyle \frac{7n\pi }{8n}}\right)$$=co{s}^2\left({\displaystyle \frac{7\pi }{8}}\right)$